Semiconductor introduction

WEBVTT
Kind: captions
Language: en

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This is Sal here and I'm with Ben Eater, who I consider to be an expert at many things

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What are we gonna talk about today.

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Hey let's talk about semiconductor.  And semiconductors is the computing revolution

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in electronics, the phones in our pockets. they are all made of chips

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that have semiconductors in them

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Right right and they are made from silicon.

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Silicon and that's why we have this periodic table here

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we see silicon here with an atomic number of 14. Which means it has 14 protons

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and a neutral silicon would also have 14 electrons

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Right but what we care about is the electrons that are in the outermost shell the valence electrons

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For any any viewers who aren't familiar we have a whole series on valence electrons and I encourage to watch them

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if what ben and I are talking seems to be a bit confusing

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But a neutral silicon atom will have 14 electrons but the 4 in the outermost shell

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that are likely to react

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and so when chemist

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depict silicon or depict how it might react

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they think primarily about this 4 electrons

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but we just drew this 4 , these are 4 VALENCE electrons

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there's another 10 for a neutral silicon atom that will be sitting closer to the nucleous

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right but those dont react with other atoms as much so we dont really think about those

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so in a semiconductor you've got basically a whole crystal of silicon atoms

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se let's see if we can imagine that

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so we have a cristal here

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each of these circles represents a silicon atom

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and we are only going to think about the valence electrons

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so this silicon atom has 4 valence electrons

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so it could be 1, 2, 3 and 4 valence electrons

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so it could be bonded to a neighboring silicon

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let me do this in a ifferent color so we can keep track of its valence electrons

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to a neighboring silicon

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whose valence electrons are 1, 2, 3 and 4

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and when we see what happened here is that they formed a covalent bond

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wich allows both of these silicons to share both of these electrons

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yeah so he drew them in different colors  but both of them feel like they own both electrons

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right , and in chemistry , once again, the atoms get stable when they have 8 valence electrons

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and so for silicon does this with 4 other silicons

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it can pretend that it has 8 valence electrons

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like this one right over here

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it kinda feels like it partially owns 8 valence electrons

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even though he only contributed 4

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the other 4 comes from the neighboring silicons

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yeah and all of the silicon atoms in this crystal structure feel that way so its very stable

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and once again, what we are seeing is just a 2-dimensional slide

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this thing would be 3-dimensional

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yeah its a 2-dimentional depiction really cause the bonds are , you know , 3 dimensional

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but yeah this is just a representation

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so what we are interested in is- we are talking about computers and electronics and so we want to talk about current flow

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and so the question is , can current flow through this , is this a conductor?

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well what we just talked about , looks like everything would be very stable here it doesnt seems to be a lot of

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strong incentive , even if you were to put a current through this , and let's do that

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even if you were to put a current through that

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so if i were to put a current across this

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se there is a batery right over here

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and this is the negative side, the negative terminal

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this is the positive terminal

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and we tipically talk about current going from positive to negative

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but we know that is actually the electrons are flowing

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yeah its a little confusing unfortunally i guess

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benjamin franklin had 50% chance of getting it right and he got it wrong

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he had it wrong well he did many other good things

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so we'll excuse him this one

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we'll give him a pass . but the electrons

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the electrons want to , you know , if you have a complete circuit

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the electrons would go the other way around

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they would go to the positive terminal

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but , all of these electrons in this silicone lattice

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it looks like they are pretty happy , there is no reason for them to move

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good conductors , when we think about the transition metals

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things like copper they have a sea of electrons

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lots of free electrons moving around around

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in this metallic bonds

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in fact this wire we have over here if it was made of , say , copper it would have just a lot of free electrons

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the sea of electrons that could flow

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but here the electrons look like they are pretty happy where they are

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they dont really have a lot of incentive to move around

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right so how is this usefull

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when we are building something like a computer

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yeah , so by itself its not

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so what we do is  , we add some impurites to the silicon

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we call this process doping

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we are just adding some impurities to the silicon that let's us conduct electricity

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those are some interesting things

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can there be like any impurity?

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no , so there are actually very specific impurities

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and so one impurity is phosforous

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i see , so let's take a look to the periodic table

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any time I hear the name of an element i need to refer to the periodic table

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to get my bearings , ALRIGHT , so phosphorous

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look is right next to silicon in the periodic table

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it has an atomic number of 15

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and so a neutral phosphorous would have 15 protons and 15 electrons

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but if we think about its valence electrons

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it would have one more

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so it has 5

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it would have 5 valence electrons

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yeah and that 5th valence electron is like super important

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so let's think about what's going on here

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if i replace this silicon with a phosphorous (blue)

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if i were to replace this silicon with phosphorous

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it could have 1, 2, 3, 4 but then it  has a 5th valence electron

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that wouldnt be able to find a covalent bond

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so he is kind of sitting there

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and let me do that again maybe this one over here is a phosphorous

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and so this is 1, 2 , 3, 4, and the it has a 5th valence electrons

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so it has this extra electron

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yeah so it has an extra electrons

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and they are free to move around , they are not stuck in these covalent bonds

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right , this covalent bond is kind of a stable configuraton

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you have the 8 valence electrons

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but here you have these extra ones , and yeah these are more , liberated i guess to move around

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AND we have this battery hooked up here , you have this positive terminal

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and electrons are gonna be atracted to that

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yeah so this one , this one right ove here it might be able to be atracted to that one , that one can move there

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and eventually get to the wire

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like this guy can replace that guy

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and eventually the electrons could start flowing

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the way we are looking at this , with two atoms could kind of make that leap

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right right , it could make the leap , it could go straight

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so this stuff could start flowing

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if this extra phosphorous electron were to move to the right

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then this phosphorous would have a positive charge

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and so maybe it grabs an eletron on from here and that grabs an electron from the wire  a

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and so you start having current flowing

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fascinating

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thats one way of doping silicon to get current flowing

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but there is another way

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and that is with something like , Boron

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let's get another lattice here

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alright so let me go back to the periodic table

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you just said boron

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so boron has an atomic number - let me go back and pick another color , purple for boron

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so boron has an atomic number of 5

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but it has 3 valence electrons

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it has 3 electrons in his outermost shell

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so boron's valence electrons may look like that depending on the convention you use

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so let's dope  this silicon lattice with a little bit of boron

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so the ones that are not labeled , lets assume they are silicon

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but i'll throw a couple of borons here

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so this is a boron

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the boron would have 3 valence electrons

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let me get rid of this one 'cause that one would've only been there if it was a silicon

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so let me change the color-im actually getting rid of this one

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because boron only has 3 valence electrons its not able to fomr this fourth covalent bond

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let me do a couple more borons

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so maybe this is a boron right over here and it has 3 valence electrons

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and so it wasnt able to form this fourth covalent bond

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and so it almost feels like there is a hole here

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yeah , and thats actually how its called

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its a hole

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that should go into "naming things"

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very creative

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sorry , yeah so this is called "holes"

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those are called holes

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they are places where an electron would want to be in a normal silicon crystal

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but there isnt one there because we've introduced this boron impurity

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i see, and i think i see were this is going becau-you know just having this extra valence electron

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introduce a kind of inbalance

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and allow electrons to flow

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it made it a little bit less stable

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eyup

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this is also a bit of an unbalance

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a little less stable , yeah

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so let'ss hook a battery up to it

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yeah let's put a battery on this

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and we get-so this is a positive , this is the negative

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so let's see , if this is a positive-this is a positive

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so electrons would want to go here

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electrons are going to be atracted to the positive terminal

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but in this case maybe we dont have a lot of free electrons

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we actually have more holes than electrons

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we have more holes than electrons so what can happen

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is that the electrons that are coming from the negative terminal

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they could , you know , they could make their way

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or , you know , they could make a chain

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but eventually these electrons can fill this hole

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yes , a nearby hole , and then maybe

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, you know , the hole will kind of transition to the - you know , one way to think about it

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is if this electron here

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fills this hole

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then you have a hole here

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yeah , just like the electrons can wonder around when we had the phosphorous over at the left

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the holes are kind of free to wonder around a bit

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right , those kind of give a place for the electrons to jump to

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exactly , yeah so electron coming out of the negative terminal of the battery

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are going to find a nearby hole

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and when they do that , then the boron

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, you know , where that hole is filled is going to be a little more positivelly charged

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a little bit more negativelly charged

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so then , a hole is gonna want to free-up

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so other electron can move

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so what will eventually happen is that , just how we depicted

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this electron could go there and then a hole would be here

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and then , maybe that electron could get to the wire over here

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and then the electrons would move to the right

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the electrons would move to the right across the wire

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and form a current

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while the holes kind of move to the left

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and we kind of see the holes as a positive thing

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even though its a "non-thing"

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yeah we like to think of the holes as moving even though its an absense of something

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but yeah we think of the holes as moving

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so , in both of these cases  , in one case we have electrons moving and in the other case

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we have holes moving

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we have a term for this

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they are called "charge carriers"

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so either the electrons or the holes are both considered charge carriers

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becuase they allow a charge to be carried  through this material

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right , they both create a kind of an unbalance

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the phosphorous creates extra electrons

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this creates holes that can be filled by other electrons

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and then the holes can keep moving

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so in the case of the left we call this N-tipe material

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that makes sense , im guessing N is for negative carrier

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yeah because the electrons are negative

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and then the boron dope thing on the right we call P-type

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and these are the two types of silicon that we use in semi-conductors

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fascinating , i think that i can already see where this can go


Liter intuition

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What I want
to talk about in this video

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is the liter.

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The liter. Let me write this down.

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The liter and the liter

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is a unit for measuring fluid volumes,

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or how much space does a fluid take up.

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So what does one liter look like?

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Well one of these milk bottles

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is a liter and your most typically

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going to think about a
liter in your everyday life

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when your buying some type
of beverages, milk, or water.

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In fact next time you
go to the grocery store

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see a lot of the bottles you get.

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They might be exactly one
liter, some of the soda bottles

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are two liters, but this
right over here is one.

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One liter, we write a capital
L for short, that's one liter

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and so there's six liters in total

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in this little carrying case

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but each of these milk
bottles is one liter.

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So that's a liter but
what if we want to go

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to really small volumes of water?

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Well if we want to go to
really small volumes of water

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we normally talk about milliliters.

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Milliliters, let me write this down.

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Milli

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Milli

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Milliliters. Actually
let me write it this way.

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Let me write milli in another color.

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Milli

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Milliliters.

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So it's a milli liter.

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And a milliliter, a milli is a prefix

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that you'll see a lot
when your using units

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and that means one
one-thousandth of a liter.

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So this means one one-thousandth.

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One one-thousandth of

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Of a

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of a liter

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and the notation for milliliter

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is a lowercase m and and uppercase L.

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Milli liter.

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Another way to think
about it, it would take

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one thousand milliliters
to get to one liter.

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So what does a milliliter look like?

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Well if you imagine
these medicine droppers,

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a milliliter would be about that high

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on one of those medicine droppers

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and a lot of these medicine droppers

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actually will have a milliliter,

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actually they'll have
up to five milliliters

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marked off but a milliliter
will be about that much

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or if you were to take a teaspoon

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from your cabinet in your kitchen

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and you would scoop out
a teaspoon of water,

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that, so this one over here
is going to be one milliliter

00:02:12.769 --> 00:02:16.936
and a teaspoon of fluid,
that's going to be about five.

00:02:18.463 --> 00:02:20.661
That's going to be about five milliliters.

00:02:20.661 --> 00:02:21.973
So another way to think about it is

00:02:21.973 --> 00:02:24.198
you would have to take a
thousand of these drops

00:02:24.198 --> 00:02:27.281
to fill one of these bottles of milk.

00:02:28.498 --> 00:02:32.093
A thousand milliliters to get to one liter

00:02:32.093 --> 00:02:34.632
or you would have to do, let's see, five,

00:02:34.632 --> 00:02:36.156
you'd have to put five,

00:02:36.156 --> 00:02:39.041
sorry, you'd have to do
two hundred teaspoons

00:02:39.041 --> 00:02:42.434
of say milk, to fill up this entire thing

00:02:42.434 --> 00:02:44.940
because each of them is five milliliters.

00:02:44.940 --> 00:02:46.544
Two hundred times five milliliters,

00:02:46.544 --> 00:02:48.164
you're going to have
a thousand milliliters

00:02:48.164 --> 00:02:49.627
which is one liter.

00:02:49.627 --> 00:02:51.435
Let me write this down.

00:02:51.435 --> 00:02:54.454
This right over here, this is one liter

00:02:54.454 --> 00:02:57.491
which is the same thing as one thousand,

00:02:57.491 --> 00:03:00.375
one thousand milliliters.

00:03:00.375 --> 00:03:02.036
Now what if we go to

00:03:02.036 --> 00:03:03.863
things that are much larger.

00:03:03.863 --> 00:03:05.393
What if we go to things
that are much larger

00:03:05.393 --> 00:03:06.951
than the scale

00:03:06.951 --> 00:03:09.059
of a jar of milk.

00:03:09.059 --> 00:03:11.311
Say a swimming pool and
this right over here,

00:03:11.311 --> 00:03:14.075
this is a picture of a
olympic size swimming pool

00:03:14.075 --> 00:03:16.443
which is, for the most
part, the largest pools,

00:03:16.443 --> 00:03:18.762
well there some larger pools
in some fancy hotel and things,

00:03:18.762 --> 00:03:20.398
but they're very large pools.

00:03:20.398 --> 00:03:21.691
So what do you think would be

00:03:21.691 --> 00:03:23.966
if one milk bottle

00:03:23.966 --> 00:03:26.042
has a, one like this,

00:03:26.042 --> 00:03:28.006
has a fluid volume of one liter,

00:03:28.006 --> 00:03:30.305
what do you think about
a whole swimming pool?

00:03:30.305 --> 00:03:32.546
Well actually a swimming like this,

00:03:32.546 --> 00:03:36.758
if it is 25 meters wide and 50 meters long

00:03:36.758 --> 00:03:39.199
and two meters deep,
it actually would have

00:03:39.199 --> 00:03:41.567
a fluid volume, are you ready for this?

00:03:41.567 --> 00:03:45.734
It would have a fluid volume
of 2.5 million liters.

00:03:49.462 --> 00:03:52.805
So if you filled this up with
milk, you can use that milk

00:03:52.805 --> 00:03:56.430
to fill up 2.5 million of these bottles.

00:03:56.430 --> 00:03:59.304
So that's just an appreciation
for just how much fluid

00:03:59.304 --> 00:04:00.396
it would take to fill up at least

00:04:00.396 --> 00:04:01.577
a big swimming pool like this,

00:04:01.577 --> 00:04:03.354
an olympic size swimming pool.

00:04:03.354 --> 00:04:05.544
A swimming pool that you
might more typically see

00:04:05.544 --> 00:04:07.787
at someone's house, who isn't, you know,

00:04:07.787 --> 00:04:09.316
that doesn't have an
olympic size swimming pool

00:04:09.316 --> 00:04:11.474
so kinda of a house size swimming pool,

00:04:11.474 --> 00:04:13.959
it's still going to be,
it's not usually to see

00:04:13.959 --> 00:04:16.734
one that will be more than one hundred,

00:04:16.734 --> 00:04:17.567
100,000

00:04:18.843 --> 00:04:20.093
100,000 liters.

00:04:21.076 --> 00:04:23.544
So once again, are you
thinking about teaspoons

00:04:23.544 --> 00:04:26.033
or drops from a dropper,

00:04:26.033 --> 00:04:27.937
your thinking in the milliliter range.

00:04:27.937 --> 00:04:29.423
If you're thinking about the type of,

00:04:29.423 --> 00:04:30.972
something that you might drink in a day

00:04:30.972 --> 00:04:33.321
or that your family might
drink over, say a week,

00:04:33.321 --> 00:04:35.320
you're thinking in,
really, the liter range.

00:04:35.320 --> 00:04:37.917
And if you're thinking about
how much fluid it would take

00:04:37.917 --> 00:04:40.507
to fill a swimming
pool, now you're talking

00:04:40.507 --> 00:04:42.716
in the tens of thousands
or hundreds of thousands

00:04:42.716 --> 00:04:45.383
or even millions of liter range.


Describing subsets of sample spaces exercise

WEBVTT
Kind: captions
Language: en

00:00:00.187 --> 00:00:02.524
- So, this right over
here is a screenshot of

00:00:02.524 --> 00:00:06.201
the Describing Subsets
of Sample Spaces exercise

00:00:06.201 --> 00:00:08.334
on Khan Academy, and
I thought I would do a

00:00:08.334 --> 00:00:10.366
couple of examples, just
because it's good practice

00:00:10.366 --> 00:00:14.368
just thinking about how do
we describe sets and subsets.

00:00:14.368 --> 00:00:17.660
So it reads, Harry Potter
is at Ollivanders Wand Shop.

00:00:17.660 --> 00:00:21.032
As we all know, the wand
must choose the wizard,

00:00:21.032 --> 00:00:23.531
so Harry cannot make the choice himself.

00:00:23.531 --> 00:00:27.209
He interprets the wand
selection as a random process

00:00:27.209 --> 00:00:30.561
so he can compare the probabilities
of different outcomes.

00:00:30.561 --> 00:00:31.943
The wood types available are

00:00:31.943 --> 00:00:34.584
holly, elm, maple, and wenge.

00:00:35.337 --> 00:00:37.509
The core materials on offer are

00:00:37.509 --> 00:00:41.390
phoenix feather, unicorn
hair, dragon scale,

00:00:41.390 --> 00:00:43.576
raven feather and thestral tail.

00:00:43.576 --> 00:00:44.738
All right!

00:00:44.738 --> 00:00:48.374
Based on the sample space of
possible outcomes listed below,

00:00:49.035 --> 00:00:50.939
what is more likely?

00:00:50.939 --> 00:00:53.672
And so, we see here, we have
four different types of woods

00:00:53.672 --> 00:00:55.937
for the wand, and then each of those

00:00:55.937 --> 00:00:58.842
could be combined with five
different types of core,

00:00:58.842 --> 00:01:01.260
Phoenix Feather, Unicorn
Hair, Dragon Scale,

00:01:01.260 --> 00:01:04.489
Raven Feather and Thestral Tail.

00:01:04.489 --> 00:01:07.132
And so, that gives us four different woods

00:01:07.132 --> 00:01:09.549
and each of those can be combined
with five different cores.

00:01:09.549 --> 00:01:11.584
20 possible outcomes.

00:01:11.584 --> 00:01:13.084
And they don't say it here but

00:01:13.084 --> 00:01:14.852
they way they're talking I guess we can,

00:01:14.852 --> 00:01:16.531
I'm going to go with the assumption that

00:01:16.531 --> 00:01:17.855
they're equally likely outcomes,

00:01:17.855 --> 00:01:19.655
although it would have been
nice if they said that,

00:01:19.655 --> 00:01:22.199
"These are all equally likely"

00:01:22.199 --> 00:01:23.931
but these are the 20 outcomes.

00:01:23.931 --> 00:01:26.514
And so, which of these are more likely?

00:01:26.514 --> 00:01:27.875
The wand that selects Harry

00:01:27.875 --> 00:01:30.943
will be made of holly or unicorn hair.

00:01:30.943 --> 00:01:32.853
So, how many of those
outcomes involve this?

00:01:32.853 --> 00:01:35.900
So if, Holly are these five outcomes

00:01:35.900 --> 00:01:38.440
and if you said, "Holly or Unicorn Hair"

00:01:38.440 --> 00:01:40.606
it's going to be these five outcomes plus,

00:01:40.606 --> 00:01:41.999
well this one involves Unicorn Hair

00:01:41.999 --> 00:01:44.555
but we've already included this one,

00:01:44.555 --> 00:01:46.892
but the other ones that's
not included for the Holly,

00:01:46.892 --> 00:01:49.432
that involve Unicorn
Hair, are the Elm Unicorn,

00:01:49.432 --> 00:01:51.545
the Maple Unicorn and the Wenge Unicorn.

00:01:51.545 --> 00:01:54.470
So it's these five, plus
these three, right over here.

00:01:54.470 --> 00:01:57.294
So eight of these 20 outcomes.

00:01:57.294 --> 00:01:59.651
And if these are all
equally likely outcomes,

00:01:59.651 --> 00:02:04.651
that means there is an
8/20 probability of a wand

00:02:04.669 --> 00:02:08.692
that will be made of
Holly or Unicorn Hair.

00:02:08.692 --> 00:02:10.460
So this is 8/20 or, that's the same

00:02:10.460 --> 00:02:12.871
thing as 4/10 or 40% chance.

00:02:12.871 --> 00:02:14.248
Now, the wand that selects Harry

00:02:14.248 --> 00:02:16.786
will be made of Holly and Unicorn Hair.

00:02:16.786 --> 00:02:18.039
Well, Holly and Unicorn Hair,

00:02:18.039 --> 00:02:20.561
that's only one out of the 20 outcomes.

00:02:20.561 --> 00:02:22.386
So this, or course, is going
to be a higher probability.

00:02:22.386 --> 00:02:24.925
It actually includes this outcome

00:02:24.925 --> 00:02:26.807
and then seven other outcomes.

00:02:26.807 --> 00:02:30.634
So, the first choice includes the outcome

00:02:30.634 --> 00:02:34.129
for the second choice
plus seven other outcomes.

00:02:34.129 --> 00:02:37.825
So this is definitely going
to be a higher probability.

00:02:37.825 --> 00:02:39.473
Let's do a couple more of these

00:02:39.473 --> 00:02:42.231
or at least one more of these.

00:02:42.601 --> 00:02:46.482
You and a friend are
playing "Fire-Water-Sponge".

00:02:46.482 --> 00:02:47.963
I've never played that game.

00:02:47.963 --> 00:02:49.895
In this game, each of the two players

00:02:49.895 --> 00:02:52.800
chooses fire, water or sponge.

00:02:52.800 --> 00:02:55.726
Both players reveal their
choice at the same time

00:02:55.726 --> 00:02:57.724
and the winner is determined
based on the choices.

00:02:57.724 --> 00:03:00.379
I guess this is like
Rock, Paper, Scissors.

00:03:00.379 --> 00:03:03.264
Fire beats sponge by burning it.

00:03:03.264 --> 00:03:05.864
Sponge beats water by soaking it up.

00:03:05.864 --> 00:03:08.425
And water beats fire by putting it out.

00:03:08.425 --> 00:03:10.260
Alright, well, it kind of makes sense.

00:03:10.260 --> 00:03:13.808
If both players choose the
same object, it is a tie.

00:03:13.808 --> 00:03:17.201
All the possible outcomes of
the game are listed below.

00:03:17.201 --> 00:03:19.720
If we take outcomes
one, three, four, five,

00:03:19.720 --> 00:03:22.402
seven and eight, as a
subset of the sample space,

00:03:22.402 --> 00:03:27.197
which of the statements
below describe the subset?

00:03:27.197 --> 00:03:30.854
So let's look at the outcomes
that they have over here.

00:03:30.854 --> 00:03:32.160
Well, it makes sense that there are

00:03:32.160 --> 00:03:34.227
nine possible outcomes because,

00:03:34.227 --> 00:03:36.563
for each of the three choices I can make

00:03:36.563 --> 00:03:39.323
there's going to be three
choices that my friend can make.

00:03:39.323 --> 00:03:41.642
So, three times three is nine.

00:03:41.642 --> 00:03:44.304
They've highlighted these red outcomes.

00:03:44.304 --> 00:03:47.351
Outcome one, three, four,
five, seven and eight.

00:03:47.351 --> 00:03:49.038
So let's see what's common about them.

00:03:49.038 --> 00:03:53.504
Outcome one, Fire, I get
Fire, friend get's Water.

00:03:53.504 --> 00:03:56.271
OK, so let's see, my friend would win.

00:03:56.271 --> 00:04:00.619
Outcome three, I pick Fire,
my friend does sponge,

00:04:00.619 --> 00:04:04.133
so actually I would win that one.

00:04:04.133 --> 00:04:07.790
And then outcome four, Water, Fire.

00:04:07.790 --> 00:04:11.248
And then outcome five, Water, Sponge.

00:04:12.202 --> 00:04:13.837
I don't see a pattern just yet,.

00:04:13.837 --> 00:04:15.687
Let's see what the choice is.

00:04:15.687 --> 00:04:18.355
The subset consists of all outcomes where

00:04:18.355 --> 00:04:20.529
your friend does not win.

00:04:20.529 --> 00:04:24.206
All outcomes where your
friend does not win.

00:04:24.206 --> 00:04:25.418
Well, that's not true because look,

00:04:25.418 --> 00:04:28.351
outcome one my friend wins.

00:04:28.351 --> 00:04:30.764
Water puts out fire so,

00:04:30.764 --> 00:04:32.879
we're not going to
select this first choice.

00:04:32.879 --> 00:04:34.615
So let's see, the subset consists of

00:04:34.615 --> 00:04:37.514
all the outcomes where your friend wins

00:04:37.514 --> 00:04:39.099
or there is a tie.

00:04:39.099 --> 00:04:40.260
So let's see.

00:04:40.260 --> 00:04:41.595
Where the friend wins or there's a tie.

00:04:41.595 --> 00:04:46.595
Well, outcome three, this is
an outcome where I would win.

00:04:47.266 --> 00:04:49.847
Or you, or whoever "Your" is,

00:04:49.847 --> 00:04:51.321
whoever they're talking about.

00:04:51.321 --> 00:04:53.045
This is one where the friend doesn't win

00:04:53.045 --> 00:04:54.577
because Fire burns Sponge so,

00:04:54.577 --> 00:04:56.632
I'm not going to select that one either.

00:04:56.632 --> 00:04:58.319
Choice three, the subset consists of

00:04:58.319 --> 00:05:02.260
all of the outcomes where
you win or there is a tie.

00:05:02.260 --> 00:05:04.007
Well, we just said, "Outcome one,

00:05:04.007 --> 00:05:05.714
I don't win that, my friend wins that.

00:05:05.714 --> 00:05:09.290
Water puts out the Fire."

00:05:09.290 --> 00:05:10.936
Now let's look at the last choice.

00:05:10.936 --> 00:05:13.109
The subset consists of all the outcomes

00:05:13.109 --> 00:05:15.852
where there is not a tie.

00:05:15.852 --> 00:05:17.925
Alright, so this is
interesting because look,

00:05:17.925 --> 00:05:20.180
outcome two there is a tie.

00:05:20.180 --> 00:05:22.374
Outcome six, there is a tie.

00:05:22.374 --> 00:05:23.963
Outcome nine, there is a tie.

00:05:23.963 --> 00:05:26.945
There's actually only three
scenarios where there's a tie.

00:05:26.945 --> 00:05:28.065
Either it's Fire, Fire.

00:05:28.065 --> 00:05:28.780
Water, Water.

00:05:28.780 --> 00:05:30.194
Or Sponge, Sponge.

00:05:30.194 --> 00:05:31.646
And those are the ones
that are not selected.

00:05:31.646 --> 00:05:33.808
So, all of these, someone is going to win.

00:05:33.808 --> 00:05:37.002
Outcome one, three, four,
five, seven or eight.

00:05:37.002 --> 00:05:41.269
So, definitely, definitely
go with that one.


Vector components from initial and terminal points

WEBVTT
Kind: captions
Language: en

00:00:00.324 --> 00:00:03.422
We can never have too much practice
getting familiar with vectors.

00:00:03.422 --> 00:00:05.711
What I want to do
in this video in particular,

00:00:05.711 --> 00:00:08.991
be able to relate
the x and y components of a vector

00:00:09.006 --> 00:00:11.616
to the initial and terminal points
to where we place it

00:00:11.616 --> 00:00:13.141
in the coordinate planes.

00:00:13.141 --> 00:00:14.488
What am I talking about?

00:00:14.488 --> 00:00:21.048
Let's say we have some vector
where its initial point is here at point a

00:00:21.051 --> 00:00:28.081
and its terminal point
is here at point b.

00:00:30.850 --> 00:00:37.960
The vector will look like this
in green.

00:00:40.519 --> 00:00:48.079
So that's what the vector looks like.

00:00:58.861 --> 00:01:02.511
It has an initial point at a
and the terminal point at b.

00:01:02.511 --> 00:01:05.388
What I want to do is,
if I know the coordinates of a and b

00:01:05.388 --> 00:01:11.968
figure out the x and y components
of my vector v.

00:01:14.953 --> 00:01:19.371
So what are the coordinates of a?

00:01:19.371 --> 00:01:25.981
The x coordinate is
-1,-2,-3,-4,-5,-6,-7,-8.

00:01:27.016 --> 00:01:30.205
So a = (-8,

00:01:30.205 --> 00:01:36.945
and the y coordinate is
1,2,3,4,5,6,7.

00:01:37.180 --> 00:01:42.550
So a=(-8, 7)

00:01:42.569 --> 00:01:47.319
and point b, the terminal point
of the vector, has the coordinates...

00:01:47.319 --> 00:01:52.961
1,2,3,4,5,6,7.
b=(7,

00:01:56.135 --> 00:02:00.248
and then the y coordinate is
-1,-2.

00:02:00.248 --> 00:02:02.380
b=(7, -2).

00:02:02.380 --> 00:02:05.386
So, given that, what are going to be
the x and y components

00:02:05.386 --> 00:02:06.882
of our vector v ?

00:02:06.882 --> 00:02:10.536
Well, let's think about it,
our vector v.

00:02:10.536 --> 00:02:14.700
There are multiple ways
that we can express a vector,

00:02:14.700 --> 00:02:20.050
one way is to use
the unit vector notation,

00:02:20.071 --> 00:02:25.304
where whatever is in parentheses
is our x component,

00:02:25.304 --> 00:02:29.614
and whatever is in these parentheses
is our y component.

00:02:29.614 --> 00:02:33.913
I'm really just scaling up the unit vector
in the horizontal direction

00:02:33.913 --> 00:02:36.675
and the unit vector
in the vertical direction.

00:02:36.675 --> 00:02:39.683
You could look at the videos
on unit vector notation

00:02:39.683 --> 00:02:41.955
if this looks foreign to you.

00:02:41.955 --> 00:02:44.870
But let's figure out
what the x component is going to be.

00:02:44.870 --> 00:02:47.545
One way to think about it is,
you could just view it as the change in x

00:02:47.545 --> 00:02:49.891
between the initial point
and the terminal point.

00:02:49.891 --> 00:02:52.593
So the change in x is going to be

00:02:52.593 --> 00:02:59.993
the terminal x, 7 minus the initial x, -8

00:03:00.009 --> 00:03:03.854
and our y component
you can just view as a change in y

00:03:03.854 --> 00:03:07.133
from the initial point
to the terminal point.

00:03:07.133 --> 00:03:10.871
So we finish at the terminal point,

00:03:10.871 --> 00:03:18.001
that is -2 minus our initial y, -7.

00:03:18.959 --> 00:03:24.062
And so what does this give us?

00:03:24.062 --> 00:03:29.532
7 minus -8,
that's the same as 7 plus 8,

00:03:31.921 --> 00:03:35.652
that's equal to 15.

00:03:35.652 --> 00:03:43.062
So 15 times i
plus -2 minus 7, that's -9,

00:03:54.204 --> 00:03:59.854
or -9j.
That's one way to write our vector.

00:03:59.854 --> 00:04:04.131
Another way is to just remember
there are a bunch of different notations

00:04:04.131 --> 00:04:05.855
people use for vectors.

00:04:05.855 --> 00:04:12.095
You could write it like this;
you could just say it's 15, -9.

00:04:18.425 --> 00:04:21.112
These are different representations
of the same vector.

00:04:21.112 --> 00:04:25.688
You could even write it like this,
vector v, you could write it as

00:04:25.688 --> 00:04:32.029
a column vector, 15, -9.

00:04:32.029 --> 00:04:34.989
So depending on what book or notation
or class you're in,

00:04:34.989 --> 00:04:37.111
you might see all of these things,

00:04:37.111 --> 00:04:39.716
but these are all the same vectors
that you know and love.

00:04:39.716 --> 00:04:44.025
We can see that over here,

00:04:44.025 --> 00:04:48.340
15 i is going to be a vector that's
15 times bigger than the unit vector

00:04:48.340 --> 00:04:50.889
in the horizontal direction.

00:04:50.889 --> 00:04:58.039
So 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15.

00:05:05.009 --> 00:05:10.079
So this is the vector 15 i.

00:05:16.313 --> 00:05:21.256
And the vector -9 j
is going to go down 9,

00:05:21.256 --> 00:05:25.806
so 1,2,3,4,5,6,7,8,9.

00:05:25.808 --> 00:05:30.978
So -9 j is going to look like this.

00:05:41.570 --> 00:05:45.185
And you see if you take 15 i
plus -9 j,

00:05:45.185 --> 00:05:48.685
you're going to get
our entire vector v,

00:05:48.685 --> 00:05:50.479
which is exactly what we wrote here.

00:05:50.479 --> 00:05:54.259
You could say the x component is 15,
the y component is -9.

00:05:54.259 --> 00:05:57.413
Or another way to say it is,
to go from point a to point b,

00:05:57.413 --> 00:05:59.779
your change in x,
you increased 15,

00:05:59.779 --> 00:06:02.266
your change in y,
you decreased 9.

00:06:02.266 --> 00:06:04.575
That tells you the x and y components

00:06:04.575 --> 00:06:07.120
of the vector that starts here
and ends there.

00:06:07.120 --> 00:06:11.654
This just happens to be
where we placed this vector,

00:06:11.654 --> 00:06:14.935
we might choose to shift them around,
but the important thing is

00:06:14.935 --> 00:06:19.755
that they keep the same
magnitude and direction.


Figuring out vector initial point

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.667
Let's say we have the vector a
and we know its x and y components.

00:00:04.679 --> 00:00:06.629
I'll write it first as a column vector.

00:00:06.629 --> 00:00:11.599
It's x component is -10
and its y component is +7.

00:00:11.607 --> 00:00:14.344
And with the many ways we've seen
to represent a vector

00:00:14.344 --> 00:00:18.701
we could also represent it
in unit vector form,

00:00:18.701 --> 00:00:20.735
so we could write it as -10

00:00:20.735 --> 00:00:23.641
times the unit vector
in the horizontal direction

00:00:23.641 --> 00:00:27.565
plus 7 times the unit vector
in the vertical direction.

00:00:27.565 --> 00:00:29.474
But either way
we could draw this thing,

00:00:29.474 --> 00:00:31.771
we could pick a place,
an initial point.

00:00:31.771 --> 00:00:33.956
If we say the initial point is the origin,

00:00:33.956 --> 00:00:35.731
we might as well
pick the origin,

00:00:35.731 --> 00:00:37.352
what would this vector look like?

00:00:37.352 --> 00:00:40.382
Well, the x component is -10,
so we'd go to -10,

00:00:40.382 --> 00:00:41.933
and then the y component is 7.

00:00:41.933 --> 00:00:44.913
So we do 10 to the left and 7 up.

00:00:44.913 --> 00:00:50.110
7 up would get us to right over there,
and the vector would look like this.

00:00:53.191 --> 00:00:56.571
So it would start at the origin
and would end,

00:00:56.586 --> 00:01:01.736
its initial point would be the origin
and its terminal point

00:01:01.753 --> 00:01:06.151
would be at x= -10 and y= 7.

00:01:06.151 --> 00:01:08.759
So this is one way to represent
the vector a.

00:01:08.759 --> 00:01:10.930
But let's say we wanted to
move it around,

00:01:10.930 --> 00:01:13.195
we wanted to have
a different terminal point.

00:01:13.195 --> 00:01:14.501
Let's say we were told

00:01:14.501 --> 00:01:18.211
instead of having
the terminal point -10, 7

00:01:18.211 --> 00:01:22.988
let's say instead
it had the terminal point 3,5.

00:01:26.476 --> 00:01:29.016
Let's make this a negative 3.

00:01:29.023 --> 00:01:31.620
-3, 5, let's say it had that
as its terminal point.

00:01:31.620 --> 00:01:34.498
So we could imagine
what the vector would look like.

00:01:34.498 --> 00:01:39.998
We would just translate this vector over
so it would look something like this.

00:01:42.072 --> 00:01:44.013
So it would terminate right over there,

00:01:44.013 --> 00:01:47.113
and it would start somewhere
in this area here.

00:01:47.113 --> 00:01:51.161
I don't know precisely,
if I could just copy/paste this

00:01:51.161 --> 00:01:54.292
and translate it over,
I could figure it out on the graph.

00:01:54.292 --> 00:01:59.882
But instead of doing it like that
let's do it mathematically.

00:01:59.909 --> 00:02:04.809
What would be the initial point
of this vector, of vector a

00:02:04.809 --> 00:02:09.969
if the terminal point is -3, 5?

00:02:12.190 --> 00:02:16.950
Let's say the initial point I
is equal to x Initial, y Initial.

00:02:21.176 --> 00:02:25.466
So x sub i, y sub i.

00:02:25.466 --> 00:02:26.661
So let's say that's it.

00:02:26.661 --> 00:02:30.341
How do we go from having
an initial point and a terminal point

00:02:30.349 --> 00:02:33.069
to getting the components of the vector?

00:02:33.069 --> 00:02:36.274
The x component of the vector
is going to be the change in x

00:02:36.274 --> 00:02:38.750
going from the initial
to the terminal point

00:02:38.750 --> 00:02:41.160
and your y component
is going to be your change in y

00:02:41.160 --> 00:02:47.092
going from your initial point
to your terminal point.

00:02:48.502 --> 00:02:53.072
So -10, your x component
is going to be your change in x

00:02:53.072 --> 00:02:55.261
going from your initial
to your terminal.

00:02:55.261 --> 00:03:00.384
That's going to be your terminal x, -3,

00:03:00.384 --> 00:03:05.156
minus your initial x sub i.

00:03:05.156 --> 00:03:08.351
Then we can do the same thing
for the y component.

00:03:08.351 --> 00:03:11.323
The y component is going to be equal to
your change in y.

00:03:11.323 --> 00:03:16.164
It's going to be your terminal y,
it's going to be 5,

00:03:16.164 --> 00:03:19.839
minus your initial y.

00:03:19.839 --> 00:03:23.353
And from this we can solve for
x sub i, your initial x,

00:03:23.353 --> 00:03:24.848
and your initial y.

00:03:24.848 --> 00:03:26.528
So let's see how we can do this.

00:03:26.528 --> 00:03:29.282
If we multiplied both sides of this
times -1,

00:03:29.282 --> 00:03:34.882
we're going to get 10 is equal to
x i +3

00:03:46.412 --> 00:03:48.273
and if we subtract 3 from both sides,

00:03:48.273 --> 00:03:54.033
we're going to get x sub i
is equal to 7.

00:03:55.579 --> 00:03:58.909
x sub i is equal to 7.

00:03:58.909 --> 00:04:01.976
Now let's do the same thing over here.

00:04:01.976 --> 00:04:05.669
If we multiplied both sides of this
times -1,

00:04:05.669 --> 00:04:10.839
we're going to have -7
is equal to y sub i.

00:04:10.855 --> 00:04:15.525
Our Initial y minus 5.

00:04:16.911 --> 00:04:23.291
Add 5 to both sides,
our Initial y is equal to -2.

00:04:24.407 --> 00:04:28.086
So our initial point for our vector,
if our terminal point is going to be -3,5

00:04:28.086 --> 00:04:33.656
based on the components of the vector,
our initial point is going to be

00:04:38.221 --> 00:04:40.289
at point 7, -2.

00:04:40.289 --> 00:04:43.714
Let's see if that makes visual sense.

00:04:43.714 --> 00:04:45.947
This is x= 7,-2.

00:04:45.947 --> 00:04:50.945
So we're saying that is our initial point,

00:04:50.945 --> 00:04:56.065
so our vector is going to look like this,

00:04:58.250 --> 00:05:00.436
which makes a lot of sense.

00:05:00.436 --> 00:05:01.963
It's just the original vector we had

00:05:01.963 --> 00:05:03.848
when we had
the original point of the origin,

00:05:03.848 --> 00:05:06.308
but we've shifted it over,

00:05:06.308 --> 00:05:12.779
and shifted it to the right,
1,2,3,4,5,6,7.

00:05:12.782 --> 00:05:15.127
And you see that right over here.

00:05:15.127 --> 00:05:17.115
Hopefully now we're pretty well-equipped.

00:05:17.115 --> 00:05:19.519
If we know the initial
and terminal point of a vector

00:05:19.519 --> 00:05:21.717
we can figure out its x and y components.

00:05:21.717 --> 00:05:24.584
Or if we know the terminal points
and the x and y components,

00:05:24.584 --> 00:05:26.374
we can figure out the initial point.

00:05:26.374 --> 00:05:28.754
Or if we know the initial point
and we know the x and y components

00:05:28.754 --> 00:05:30.811
we can figure out the terminal point.

00:05:30.811 --> 00:05:32.631
So if we know two out of these three,

00:05:32.631 --> 00:05:35.502
we can figure out
the third piece of information.


Constructing a tangent line using compass and straightedge

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:03.134
You'd be amazed at the things
you can draw or construct

00:00:03.134 --> 00:00:05.674
if you have a straightedge
and a compass.

00:00:05.674 --> 00:00:08.815
A straightedge is literally
just something that has a straight edge

00:00:08.815 --> 00:00:11.286
that allows you to draw straight lines,
like a ruler.

00:00:11.286 --> 00:00:13.819
A compass is something that allows you
to draw circles

00:00:13.819 --> 00:00:16.765
centered where you want them
to be centered of different radii.

00:00:16.765 --> 00:00:18.825
Your typical compass would be
a metal thing

00:00:18.825 --> 00:00:21.657
that has a pin on one end
and is shaped like an angle

00:00:21.657 --> 00:00:23.921
and you have a pencil
at the other end.

00:00:23.921 --> 00:00:27.343
I don't have real physical
rulers and pencils in front of me

00:00:27.343 --> 00:00:29.598
but I have the virtual equivalent.

00:00:29.598 --> 00:00:32.742
I can say 'add a compass'
and draw a circle.

00:00:32.742 --> 00:00:36.889
I can pick where I want to center it
and I can change the radius,

00:00:36.889 --> 00:00:42.169
and I can draw a straight line segment
and move it around.

00:00:42.169 --> 00:00:44.220
This is equivalent
to having a straightedge.

00:00:44.220 --> 00:00:46.057
Using these tools,

00:00:46.057 --> 00:00:52.987
I want to construct a line going through P
that is tangent to the circle.

00:00:52.990 --> 00:00:56.885
I'll draw a line that looks
something like this.

00:00:56.885 --> 00:01:01.447
Remember a tangent line
will touch the circle exactly at one point

00:01:01.447 --> 00:01:03.430
and that point,
since it's going through b

00:01:03.430 --> 00:01:04.571
should be point P.

00:01:04.571 --> 00:01:06.575
Another way to think about a tangent line

00:01:06.575 --> 00:01:10.195
is that it's going to be perpendicular
to the radius between that point

00:01:10.195 --> 00:01:12.866
and the center.

00:01:12.866 --> 00:01:16.502
What I just drew looks pretty good,
but it's not so precise.

00:01:16.502 --> 00:01:19.316
I don't know if it's exactly
perpendicular to the radius,

00:01:19.316 --> 00:01:22.727
I don't know if it's touching it
exactly at one point right there.

00:01:22.727 --> 00:01:25.802
What we're going to do is
use our virtual compass

00:01:25.802 --> 00:01:30.952
and our virtual straightedge
to do a more precise drawing.

00:01:30.954 --> 00:01:33.569
Let's do that.

00:01:33.569 --> 00:01:39.050
The first thing I'm going to do is
set up P as the mid-point of a line

00:01:39.050 --> 00:01:42.763
where the center of the circle
is one other end of the line.

00:01:42.763 --> 00:01:47.793
The way I can do that--
let me 'add a compass' here.

00:01:47.793 --> 00:01:52.702
I'll construct a circle
that has the same radius

00:01:52.702 --> 00:01:54.792
as my original circle.

00:01:54.792 --> 00:01:59.790
Now let me move it over
so now it's centered at P.

00:01:59.790 --> 00:02:03.420
Why is this useful?

00:02:03.420 --> 00:02:05.699
Now a diameter of this new circle

00:02:05.699 --> 00:02:07.677
is going to be a segment
that is centered at P.

00:02:07.677 --> 00:02:11.269
I'm going to have a segment
where P is the mid-point

00:02:11.269 --> 00:02:15.919
and then the center of my original circle
is going to be one of the end points.

00:02:15.932 --> 00:02:17.257
Let's do that.

00:02:17.257 --> 00:02:20.489
I'm going to add a straightedge
and make that one of the end points

00:02:20.489 --> 00:02:22.769
and I'm going to go through P

00:02:22.769 --> 00:02:26.968
all the way to the other side
of my new circle.

00:02:26.968 --> 00:02:31.738
What was the whole point
of me doing that?

00:02:34.513 --> 00:02:37.350
Now I've made P
the mid-point of a segment

00:02:37.350 --> 00:02:40.680
so if I can construct
a perpendicular bisector

00:02:40.695 --> 00:02:42.063
of the segment

00:02:42.063 --> 00:02:44.794
it will go through P
because P is the mid-point

00:02:44.794 --> 00:02:49.608
and that thing is going to be
exactly perpendicular to the radius

00:02:49.608 --> 00:02:55.818
because the original radius
is part of this segment.

00:03:02.526 --> 00:03:04.645
Let's see how I can do this.

00:03:04.645 --> 00:03:09.217
What I could do is--
I'm going to draw another circle.

00:03:09.217 --> 00:03:13.708
I'm going to center it
at the original circle

00:03:13.708 --> 00:03:15.623
and make it have a different radius.

00:03:15.623 --> 00:03:19.620
Maybe a radius something like that.

00:03:19.620 --> 00:03:22.701
Now I'm going to construct another circle
of this larger size

00:03:22.701 --> 00:03:26.315
but I'm going to center it
at this point over here.

00:03:26.315 --> 00:03:29.085
I think you'll see quickly
what this will accomplish.

00:03:29.085 --> 00:03:35.046
So I'll construct another circle
of that same larger radius.

00:03:40.075 --> 00:03:45.405
Now I'm going to move it
over here.

00:03:46.629 --> 00:03:48.731
So what's interesting
about the intersection

00:03:48.731 --> 00:03:50.931
of these two larger circles?

00:03:50.931 --> 00:03:56.957
This point over here
is equidistant to this end of the segment

00:03:56.957 --> 00:03:58.765
and to this end of the segment.

00:03:58.765 --> 00:04:02.107
Remember these two larger circles
have the same radius.

00:04:02.107 --> 00:04:04.517
So if I'm sitting on both of them

00:04:04.517 --> 00:04:08.586
I am that distance away from this point
and that distance away from this point.

00:04:08.586 --> 00:04:13.064
So something that is equidistant
from the two end points of a segment

00:04:13.064 --> 00:04:16.105
is going to sit
on the perpendicular bisector.

00:04:16.105 --> 00:04:18.856
So this point is going to sit
on the perpendicular bisector

00:04:18.856 --> 00:04:22.413
and this point is going to be sitting
on the perpendicular bisector.

00:04:22.413 --> 00:04:25.024
So now we can draw
a perpendicular bisector.

00:04:25.024 --> 00:04:29.468
We can go from this point,
the intersection of our two larger circles

00:04:29.468 --> 00:04:33.863
this point that is equidistant
from the two centers of the large circle,

00:04:33.863 --> 00:04:37.212
to this point that is equidistant
to the two centers of the large circle.

00:04:37.212 --> 00:04:40.028
And once again, it is equidistant
to the two centers of the large circle,

00:04:40.028 --> 00:04:43.810
but those points are also
the end points of this segment.

00:04:43.810 --> 00:04:46.749
So these two points are on
the perpendicular bisector,

00:04:46.749 --> 00:04:48.492
you just need two points for a line.

00:04:48.492 --> 00:04:51.122
So I've just constructed
a perpendicular bisector to P

00:04:51.122 --> 00:04:56.612
and it's perpendicular to the radius
from the center

00:04:56.612 --> 00:04:59.448
to P of our original circle.

00:04:59.448 --> 00:05:05.818
Now, that is going to be a tangent line
because if we go through P

00:05:05.818 --> 00:05:10.605
and we are exactly perpendicular
to the radius from P to the center

00:05:10.605 --> 00:05:15.715
then this line we've just constructed
is actually tangent.

00:05:15.724 --> 00:05:18.453
So, it might seem like a lot of work
to do all of this,

00:05:18.453 --> 00:05:20.317
I could have started just eyeballing it

00:05:20.317 --> 00:05:24.727
but when we do it like this
we can feel really good

00:05:24.746 --> 00:05:26.255
that we're being precise.

00:05:26.255 --> 00:05:28.892
Imagine if you were trying to do this
on a larger scale,

00:05:28.892 --> 00:05:31.730
if you were trying to engineer
some very precise instrument,

00:05:31.730 --> 00:05:33.366
you would want to do it this way.

00:05:33.366 --> 00:05:35.540
You would want to draw
a very precise drawing,

00:05:35.540 --> 00:05:37.058
maybe an architectural drawing.

00:05:37.058 --> 00:05:39.299
This could be an interesting way
to approach it.

00:05:39.299 --> 00:05:43.493
In times past before folks
had things like computers,

00:05:43.493 --> 00:05:47.803
this was a thing people actually did.


Another example using compass and straightedge for tangent line

WEBVTT
Kind: captions
Language: en

00:00:00.357 --> 00:00:01.948
- [Voiceover] Let's do another example of

00:00:01.948 --> 00:00:04.931
using a virtual compass
and virtual straight edge

00:00:04.931 --> 00:00:08.228
to draw a tangent line to a circle.

00:00:08.228 --> 00:00:10.631
And now we're told to
construct a line going through

00:00:10.631 --> 00:00:12.210
P tangent to the circle.

00:00:12.210 --> 00:00:15.507
And in this example, P
doesn't sit on this circle,

00:00:15.507 --> 00:00:17.388
P is outside of the circle.

00:00:17.388 --> 00:00:19.466
So we wanna draw something like,

00:00:19.466 --> 00:00:22.995
and actually let me get
my straight edge out.

00:00:25.256 --> 00:00:28.165
So my straight edge, and I
have my controls up here,

00:00:28.165 --> 00:00:29.651
this is on the Kahn Academy exercise

00:00:29.651 --> 00:00:31.369
Constructing a Line Tangent to a Circle,

00:00:31.369 --> 00:00:32.391
so I have that up there.

00:00:32.391 --> 00:00:34.411
So let me add the straight edge.

00:00:34.411 --> 00:00:36.431
Once again, you could try to eyeball it.

00:00:36.431 --> 00:00:39.391
We're gonna go through P, going
to be tangent to the circle,

00:00:39.391 --> 00:00:41.852
so hey yeah, maybe something like that.

00:00:41.852 --> 00:00:43.907
That looks pretty good,
but like we've said

00:00:43.907 --> 00:00:46.484
in other geometric constructions videos,

00:00:46.484 --> 00:00:48.353
that's just eyeballing it, we don't know

00:00:48.353 --> 00:00:50.652
how precise that actually is.

00:00:50.652 --> 00:00:52.649
So what if we had a compass
and a straight edge,

00:00:52.649 --> 00:00:54.669
how do we make something more precise?

00:00:54.669 --> 00:00:55.864
And actually in the process of doing it,

00:00:55.864 --> 00:00:57.606
we'll make really fun patterns.

00:00:57.606 --> 00:00:59.405
So how do we do that?

00:00:59.405 --> 00:01:02.064
Well, what I'm gonna do is,

00:01:02.064 --> 00:01:04.443
I'm gonna try to construct a circle

00:01:04.443 --> 00:01:08.565
that has the segment PC

00:01:08.565 --> 00:01:11.397
as a diameter.

00:01:13.456 --> 00:01:15.616
So let me draw that.

00:01:15.616 --> 00:01:17.821
So I wanna construct a circle that has

00:01:17.821 --> 00:01:21.919
segment PC as a diameter.

00:01:24.071 --> 00:01:25.813
And in order to do that, I need to

00:01:25.813 --> 00:01:27.496
figure out where the
center of that circle is.

00:01:27.496 --> 00:01:29.307
And you'll see in a second
why it's useful to have

00:01:29.307 --> 00:01:31.687
this circle that has PC as a diameter.

00:01:31.687 --> 00:01:33.289
So where is the center?

00:01:33.289 --> 00:01:34.554
It looks like it's roughly here,

00:01:34.554 --> 00:01:35.936
but how do we actually construct it?

00:01:35.936 --> 00:01:36.876
To do that, we're gonna have to figure out

00:01:36.876 --> 00:01:40.986
the midpoint of segment PC, or CP.

00:01:40.986 --> 00:01:43.795
And to do that, what I'm gonna do is

00:01:43.795 --> 00:01:45.792
construct two circles.

00:01:45.792 --> 00:01:47.986
I'll just make 'em a little bit larger.

00:01:47.986 --> 00:01:50.067
So one centered at C.

00:01:51.721 --> 00:01:54.983
I'll make it reasonably
large, maybe that big.

00:01:54.983 --> 00:01:57.259
Then I'm gonna construct another circle

00:01:57.259 --> 00:01:59.277
of the same radius

00:02:00.416 --> 00:02:02.285
as that large one that I just constructed,

00:02:02.285 --> 00:02:03.737
so it's that same radius.

00:02:03.737 --> 00:02:06.174
But I'm now going to center it at P.

00:02:09.319 --> 00:02:12.059
Now why is this interesting?

00:02:12.059 --> 00:02:14.381
Why is what I just did interesting?

00:02:14.381 --> 00:02:17.155
Well where these two
larger circles intersect

00:02:17.155 --> 00:02:20.859
are going to be equidistant to P and C.

00:02:20.859 --> 00:02:22.031
How do we know that?

00:02:22.031 --> 00:02:24.933
Well all the points on this
circle are equidistant to C,

00:02:24.933 --> 00:02:27.871
all the points on this
circle are equidistant to P,

00:02:27.871 --> 00:02:30.111
and these circles have the same radius.

00:02:30.111 --> 00:02:31.899
And at this point right over here,

00:02:31.899 --> 00:02:33.373
they are equidistant to both of them,

00:02:33.373 --> 00:02:35.416
because it sits on both circles.

00:02:35.416 --> 00:02:37.715
So this point is
equidistant to both of them,

00:02:37.715 --> 00:02:39.932
and so is this point right over here.

00:02:39.932 --> 00:02:42.289
And so they both sit on the perpendicular

00:02:42.289 --> 00:02:44.529
bisector of this segment.

00:02:44.529 --> 00:02:45.969
Of the segment CP.

00:02:45.969 --> 00:02:47.675
So actually let me draw that.

00:02:47.675 --> 00:02:50.752
So if I draw a line that
looks something like this,

00:02:50.752 --> 00:02:53.550
this right over here is
a perpendicular bisector

00:02:53.550 --> 00:02:57.392
of line, of segment CP.

00:02:57.392 --> 00:02:59.714
Now what we really just care about is that

00:02:59.714 --> 00:03:01.850
it is bisecting it, 'cause we wanted

00:03:01.850 --> 00:03:03.475
to find the midpoint.

00:03:03.475 --> 00:03:05.611
And now that we have found
the midpoint of our segment,

00:03:05.611 --> 00:03:08.328
we're ready to construct
that circle I talked about.

00:03:08.328 --> 00:03:11.451
A circle centered at the midpoint,

00:03:13.390 --> 00:03:17.347
and it has CP as a diameter.

00:03:18.193 --> 00:03:20.526
So I got that far, but
why did I go through

00:03:20.526 --> 00:03:22.523
all of this trouble to do that?

00:03:22.523 --> 00:03:24.987
Well now we're going to use the idea

00:03:24.987 --> 00:03:28.470
that if you have a triangle
embedded in a circle

00:03:28.470 --> 00:03:31.210
where one side of the
triangle is a diameter,

00:03:31.210 --> 00:03:34.170
then you're going to
have a right triangle.

00:03:34.170 --> 00:03:35.830
Well what am I talking about?

00:03:35.830 --> 00:03:38.047
Let me actually just draw the triangle.

00:03:38.047 --> 00:03:40.369
So let me add a straight edge here.

00:03:40.369 --> 00:03:42.482
So I'm gonna draw a triangle,

00:03:42.482 --> 00:03:44.444
so it has one side as the diameter,

00:03:44.444 --> 00:03:46.127
so we're gonna embed it in this circle

00:03:46.127 --> 00:03:48.089
that I just constructed,
this circle centered

00:03:48.089 --> 00:03:50.307
at this midpoint, this
circle right over here.

00:03:50.307 --> 00:03:52.431
CP is clearly our diameter.

00:03:52.431 --> 00:03:54.788
And I'm gonna embed it
in CP, but I'm gonna

00:03:54.788 --> 00:03:56.366
put it in this point right over here,

00:03:56.366 --> 00:03:59.385
'cause this point sits on
the circle centered at C.

00:03:59.385 --> 00:04:02.566
And I'm gonna draw another line.

00:04:02.566 --> 00:04:07.000
So this other line, let me put
that there, and this there.

00:04:07.000 --> 00:04:10.306
So my claim is that this
is a right triangle.

00:04:10.306 --> 00:04:11.537
How do I make this claim,

00:04:11.537 --> 00:04:13.081
and I've proved it in other videos,

00:04:13.081 --> 00:04:14.717
is it's embedded in a circle,

00:04:14.717 --> 00:04:16.587
it's embedded in this
circle right over here

00:04:16.587 --> 00:04:18.479
that's highlighted in yellow.

00:04:18.479 --> 00:04:22.101
It has a diameter for one of its sides,

00:04:22.101 --> 00:04:24.086
and that side is actually its hypotenuse.

00:04:24.086 --> 00:04:26.477
So and we prove it in other videos.

00:04:26.477 --> 00:04:29.066
So this is a right triangle.

00:04:29.066 --> 00:04:31.597
Well why is this useful for proving,

00:04:31.597 --> 00:04:34.476
or why is this useful for
constructing a tangent line

00:04:34.476 --> 00:04:36.484
to this circle over here?

00:04:36.484 --> 00:04:38.760
To circle centered at C?

00:04:38.760 --> 00:04:41.256
Well, this side right over
here that I have in orange,

00:04:41.256 --> 00:04:43.078
that I'm highlighting right now,

00:04:43.078 --> 00:04:44.785
that's the radius of C.

00:04:44.785 --> 00:04:46.224
And if that forms the right angle

00:04:46.224 --> 00:04:47.733
with this line right over here,

00:04:49.305 --> 00:04:51.580
then this line right over
here must be tangent.

00:04:51.580 --> 00:04:52.811
And to really make it look tangent,

00:04:52.811 --> 00:04:54.808
I'll elongate it a little
bit just like that.

00:04:54.808 --> 00:04:55.725
And there you go.

00:04:55.725 --> 00:04:57.246
You feel good that this looks like

00:04:57.246 --> 00:04:59.521
it's truly intersecting at a right angle,

00:04:59.521 --> 00:05:01.146
and that this segment
that I'm highlighting

00:05:01.146 --> 00:05:03.921
in orange right now, is truly tangent.

00:05:03.921 --> 00:05:06.486
So once again, a lot more trouble

00:05:06.486 --> 00:05:08.669
than just eyeballing it and trying to

00:05:08.669 --> 00:05:10.550
draw it and actually,
a lot of you probably

00:05:10.550 --> 00:05:12.546
could have eyeballed this
segment right over here.

00:05:12.546 --> 00:05:14.009
But if you're doing
something on a larger scale,

00:05:14.009 --> 00:05:16.459
you wanna be more precise,
it's useful to be able

00:05:16.459 --> 00:05:17.840
to do these constructions.

00:05:17.840 --> 00:05:20.127
And frankly, while you're
in the process here,

00:05:20.127 --> 00:05:21.961
while you're dealing with the compass and

00:05:21.961 --> 00:05:23.505
the ruler or the straight edge,

00:05:23.505 --> 00:05:26.106
you get a new appreciation
for what these tools can do,

00:05:26.106 --> 00:05:27.998
and you're also making some pretty

00:05:27.998 --> 00:05:30.378
fun patterns and designs.

00:05:30.378 --> 00:05:33.046
I almost feel like hanging
this one up on my wall.


Showing angles have same measure

WEBVTT
Kind: captions
Language: en

00:00:00.212 --> 00:00:02.268
We have an interesting looking
diagram here.

00:00:02.268 --> 00:00:04.745
Let's see if we know a few things
about this diagram.

00:00:04.745 --> 00:00:11.195
Let's say we know that line MK

00:00:14.861 --> 00:00:19.781
is parallel to line NJ.

00:00:19.781 --> 00:00:23.399
So this line is parallel to this line.

00:00:23.399 --> 00:00:25.863
This is line MK, this is line NJ.

00:00:25.863 --> 00:00:28.070
Now, given that
and all the other information

00:00:28.070 --> 00:00:29.231
on this diagram,

00:00:29.231 --> 00:00:36.041
I'm hoping to prove that the measure
of this angle LMK

00:00:38.589 --> 00:00:43.105
is equal to the measure
of this angle over here

00:00:43.105 --> 00:00:49.005
and this angle is LNJ.

00:00:53.045 --> 00:00:56.102
Another way of writing this is;

00:00:56.102 --> 00:01:01.722
the measure of LMK is b
and the measure of LNK is a.

00:01:01.722 --> 00:01:04.422
So we want to prove that b is equal to a

00:01:04.422 --> 00:01:06.793
using all this information
that we know.

00:01:06.793 --> 00:01:09.764
Like always I encourage you
to try this on your own

00:01:09.764 --> 00:01:12.102
before I walk through it.

00:01:12.102 --> 00:01:14.087
Alright, let's walk through it.

00:01:14.087 --> 00:01:20.547
The first thing you might see is
I have a triangle formed up here,

00:01:23.501 --> 00:01:25.865
triangle MLK.

00:01:25.865 --> 00:01:31.315
What do we know about the measurement
of the interior angles of a triangle?

00:01:31.315 --> 00:01:33.685
The measures of the interior angles
of a triangle

00:01:33.685 --> 00:01:37.301
are going to add up to 180 degrees.

00:01:37.301 --> 00:01:41.821
We know that b,
which is the measure of this angle

00:01:41.821 --> 00:01:47.081
plus the measure of this angle, c

00:01:47.097 --> 00:01:51.867
plus the measure of this right angle,
which is plus 90 degrees

00:01:57.011 --> 00:02:03.021
is going to be equal to 180 degrees.

00:02:08.291 --> 00:02:11.852
And so if we subtract 90 degrees
from both sides

00:02:11.852 --> 00:02:16.902
we're going to get b plus c
is equal to 180 degrees minus 90 degrees.

00:02:25.423 --> 00:02:30.011
It's going to be 90 degrees.

00:02:30.011 --> 00:02:32.635
Or if we wanted to solve
explicitly for b

00:02:32.635 --> 00:02:34.873
we could subtract c from both sides,

00:02:34.873 --> 00:02:41.993
and we could write b is equal to
90 degrees minus c.

00:02:49.015 --> 00:02:53.277
So that's interesting,
that's one way of expressing b.

00:02:53.277 --> 00:02:55.504
Can we express a in a similar way?

00:02:55.504 --> 00:02:57.584
Once again, if at any point
you get inspired

00:02:57.591 --> 00:02:59.015
I encourage you to do that.

00:02:59.015 --> 00:03:05.855
If we look carefully we see that
we have triangle NLJ,

00:03:12.934 --> 00:03:17.446
this really big triangle,
it's really most of the diagram.

00:03:17.446 --> 00:03:23.466
What's interesting about NLJ
is that J is another right triangle,

00:03:23.476 --> 00:03:27.059
c is one of the measures
of one of the interior angles,

00:03:27.059 --> 00:03:29.356
and a is a measure of
the other interior angle.

00:03:29.356 --> 00:03:31.081
We can write something very similar,

00:03:31.081 --> 00:03:37.011
we can write a plus c
plus 90 degrees

00:03:45.344 --> 00:03:52.014
is going to be equal
to 180 degrees.

00:03:59.463 --> 00:04:01.398
So what can we do here?

00:04:01.398 --> 00:04:05.007
We can do the exact same process
to solve for a.

00:04:05.007 --> 00:04:08.631
If we subtract 90 from both sides
and we subtract c from both sides

00:04:08.631 --> 00:04:11.719
what do we get?

00:04:11.719 --> 00:04:18.299
We get a is equal to 90 degrees
and if you subtract c from both sides

00:04:20.725 --> 00:04:24.754
you're going to get 90 degrees minus c.

00:04:24.754 --> 00:04:26.906
Now this is interesting,

00:04:26.906 --> 00:04:30.566
b is equal to 90 degrees minus c
and a is equal to 90 degrees minus c.

00:04:30.566 --> 00:04:34.611
So 90 degrees minus c is equal to a,
it's also equal to b.

00:04:34.611 --> 00:04:41.321
Or, we can now say that
a must be equal to b,

00:04:41.329 --> 00:04:45.000
that the measure of angle LMK
which is b

00:04:45.000 --> 00:04:50.340
is equal to the measure of angle LNJ
which is equal to a.


Line and angle proofs exercise

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:02.837
- [Voiceover] This right
here is a screenshot of

00:00:02.837 --> 00:00:05.860
the line and angle proofs
exercise on Khan Academy,

00:00:05.860 --> 00:00:08.004
and I thought we would
use this to really just

00:00:08.004 --> 00:00:09.948
get some practice with
line and angle proofs.

00:00:09.948 --> 00:00:12.404
And what's neat about this, this even uses

00:00:12.404 --> 00:00:14.756
translations and transformations

00:00:14.756 --> 00:00:17.195
as ways to actually prove things.

00:00:17.195 --> 00:00:19.272
So let's look at what they're telling us.

00:00:19.272 --> 00:00:23.709
So it says line AB and
line DE are parallel lines.

00:00:23.709 --> 00:00:25.323
All right.

00:00:25.323 --> 00:00:27.668
Perform a translation that proves

00:00:27.668 --> 00:00:30.372
corresponding angles are always equal,

00:00:30.372 --> 00:00:33.676
and select the option
which explains the proof.

00:00:33.676 --> 00:00:37.163
All right, so let's see
what they have down here.

00:00:37.163 --> 00:00:39.908
So they say perform a translation

00:00:39.908 --> 00:00:43.428
that proves corresponding
angles are always equal.

00:00:43.428 --> 00:00:46.904
And then select the option
that explains the proof.

00:00:46.904 --> 00:00:49.947
So they've picked two
corresponding angles here.

00:00:49.947 --> 00:00:52.114
And so you see this is kind
of the bottom left angle,

00:00:52.114 --> 00:00:54.442
this phi, and then you
have theta right here

00:00:54.442 --> 00:00:56.170
is the bottom left angle down here,

00:00:56.170 --> 00:00:58.139
so these are corresponding angles,

00:00:58.139 --> 00:01:01.756
line FB is a transversal,
and they already told us

00:01:01.756 --> 00:01:05.316
that line AB, what did they
call it, did they call it DE?

00:01:05.316 --> 00:01:08.959
And line DE, are indeed, parallel lines.

00:01:08.959 --> 00:01:12.323
So we wanna prove that these two things,

00:01:12.323 --> 00:01:15.689
that the measure of these
two angles are equal.

00:01:15.689 --> 00:01:20.219
So, there's many ways that
you can do this geometrically.

00:01:20.219 --> 00:01:23.281
And we do that in many
Khan Academy videos,

00:01:23.281 --> 00:01:26.494
but this one, they offer us
the option of translating,

00:01:26.494 --> 00:01:28.537
of doing a translation,
so let's see what that is.

00:01:28.537 --> 00:01:30.976
So I press the translate
button, and when I move this

00:01:30.976 --> 00:01:33.213
around notice it essentially translates

00:01:33.213 --> 00:01:36.941
these four points, which has the effect of

00:01:36.941 --> 00:01:40.095
translating this entire intersection here.

00:01:40.095 --> 00:01:44.088
So if we take, this point
where my mouse is right now,

00:01:44.088 --> 00:01:47.984
that is the point D, and
I'm translating it around.

00:01:47.984 --> 00:01:51.813
If I move that over to B,

00:01:51.813 --> 00:01:54.518
what it shows is, because under

00:01:54.518 --> 00:01:57.352
a translation the angle
measures shouldn't change.

00:01:57.352 --> 00:02:00.558
So when I did that, this angle,

00:02:00.558 --> 00:02:04.757
so it's down here, theta is
the measure of angle CDF.

00:02:04.757 --> 00:02:07.359
And so when you move it over here,

00:02:07.359 --> 00:02:09.322
this right over here should be the same.

00:02:09.322 --> 00:02:12.212
This angle's measure is the same as CDF,

00:02:12.212 --> 00:02:13.829
I'm just translating it.

00:02:13.829 --> 00:02:16.125
And when you move it
over here, you see look,

00:02:16.125 --> 00:02:19.477
that's the same exact measure as phi.

00:02:19.477 --> 00:02:21.088
So this is one way to think about it.

00:02:21.088 --> 00:02:25.238
I just translated the point D to B,

00:02:25.238 --> 00:02:28.714
and then it really just
translated angle CDF

00:02:28.714 --> 00:02:31.725
over angle ABD.

00:02:31.725 --> 00:02:34.614
So to show that these
have the same measure.

00:02:34.614 --> 00:02:37.444
Or at least to feel good
about the idea of them

00:02:37.444 --> 00:02:38.926
having the same measure.

00:02:38.926 --> 00:02:41.421
So let's see which choices describe that.

00:02:41.421 --> 00:02:44.554
So let me, I'm having
trouble operating my mouse.

00:02:44.554 --> 00:02:48.144
All right, the translation mapping

00:02:48.144 --> 00:02:51.643
point F to point D, so point F

00:02:51.643 --> 00:02:55.705
to point D, point F to point D,

00:02:55.705 --> 00:02:57.937
we didn't map point F to point D.

00:02:57.937 --> 00:02:59.506
So this is already looking suspect,

00:02:59.506 --> 00:03:02.960
produces a new line which
is a bisector of segment DB.

00:03:02.960 --> 00:03:07.496
A new line which is a
bisector of segment B,

00:03:07.496 --> 00:03:10.669
okay, this doesn't seem
anything like what I just did,

00:03:10.669 --> 00:03:12.774
so I'm just gonna move on to the next one.

00:03:12.774 --> 00:03:15.357
Since the image of a
line under translation

00:03:15.357 --> 00:03:17.941
is parallel to the
original line, that's true,

00:03:17.941 --> 00:03:21.909
the translation that
maps point D to point B,

00:03:21.909 --> 00:03:23.281
that's what I did right over here,

00:03:23.281 --> 00:03:26.690
maps angle CDF to ABD.

00:03:26.690 --> 00:03:29.986
And that's what I did, I mapped angle CDF

00:03:29.986 --> 00:03:34.986
to angle ADB, that'e exactly
what I did right over there.

00:03:35.797 --> 00:03:40.190
So this is to ABD, translations
preserve angle measures,

00:03:40.190 --> 00:03:42.605
so theta is equal to phi.

00:03:42.605 --> 00:03:44.477
Yup, that one looks pretty good.

00:03:44.477 --> 00:03:46.931
The translation that mapped point D to E.

00:03:46.931 --> 00:03:49.347
I didn't do that, I didn't take

00:03:49.347 --> 00:03:52.419
point D and move it over to E like that,

00:03:52.419 --> 00:03:54.043
that didn't really help me.

00:03:54.043 --> 00:03:56.540
Let's just keep reading
it just to make sure.

00:03:56.540 --> 00:03:59.740
Produces a parallelogram,
that actually is true,

00:03:59.740 --> 00:04:03.905
if I translate point D to point E it does,

00:04:03.905 --> 00:04:07.208
I have this parallelogram constructed,

00:04:07.208 --> 00:04:09.169
but it really doesn't help us establishing

00:04:09.169 --> 00:04:11.026
that phi is equal to theta.

00:04:11.026 --> 00:04:13.912
So that one I also don't feel good about.

00:04:13.912 --> 00:04:14.928
So and it's good because

00:04:14.928 --> 00:04:16.720
we felt good about the middle choice.

00:04:16.720 --> 00:04:19.412
Let's do one more of these.

00:04:19.412 --> 00:04:24.105
So they are telling us that line AOB,

00:04:26.701 --> 00:04:29.493
and they could have just said line AB,

00:04:29.493 --> 00:04:32.040
but I guess they wanted to put
the O in there to show that

00:04:32.040 --> 00:04:35.833
point O is on that line,
that AOB are colinear.

00:04:35.833 --> 00:04:39.719
And COD is our straight
lines, all right, fair enough.

00:04:39.719 --> 00:04:41.376
Which of these statements prove

00:04:41.376 --> 00:04:43.679
vertical angles are always equal?

00:04:43.679 --> 00:04:45.679
So vertical angles would be the angles

00:04:45.679 --> 00:04:47.868
on the opposite sides of an intersection.

00:04:47.868 --> 00:04:49.704
So in order to prove that vertical,

00:04:49.704 --> 00:04:53.631
so for example angle AOC, and angle DOB,

00:04:53.631 --> 00:04:55.575
are vertical angles.

00:04:55.575 --> 00:04:58.905
And if we wanted to prove
that they are equal,

00:04:58.905 --> 00:05:00.981
we would say well their
measures are gonna be equal,

00:05:00.981 --> 00:05:03.182
so theta should be equal to phi.

00:05:03.182 --> 00:05:06.489
So let's see which of these
statements actually does that.

00:05:06.489 --> 00:05:11.423
So this one says segment
OA is congruent to OD.

00:05:11.423 --> 00:05:14.148
OA is congruent to OD.

00:05:14.148 --> 00:05:17.388
We don't know that, they
never even told us that.

00:05:17.388 --> 00:05:19.885
So I don't even have
to read the rest of it,

00:05:19.885 --> 00:05:21.823
this is already saying,
I don't know how far

00:05:21.823 --> 00:05:23.940
D is away from O, I don't know if it's

00:05:23.940 --> 00:05:26.068
the same distance A is from O.

00:05:26.068 --> 00:05:27.837
So we can just rule this first choice out.

00:05:27.837 --> 00:05:30.485
I can stop reading, this
started with a statement

00:05:30.485 --> 00:05:33.147
that we don't know, based on
the information they gave us.

00:05:33.147 --> 00:05:35.372
So let's look at the second choice.

00:05:35.372 --> 00:05:40.270
If ray OA and ray OC are each

00:05:40.270 --> 00:05:43.779
rotated 180 degrees about point O,

00:05:43.779 --> 00:05:48.460
they must map to OB and OD respectively.

00:05:48.460 --> 00:05:51.633
If two rays are rotated
by the same amount,

00:05:51.633 --> 00:05:54.057
the angle between them will not change,

00:05:54.057 --> 00:05:56.392
so phi must be equal to theta.

00:05:56.392 --> 00:05:58.080
So this is interesting, so let's just

00:05:58.080 --> 00:06:00.441
slow down and think about
what they're saying.

00:06:00.441 --> 00:06:04.192
If ray OA and OC are
each rotated 180 degrees,

00:06:04.192 --> 00:06:06.664
so if you take ray OA,
this right over here,

00:06:06.664 --> 00:06:10.666
if you rotate it 180
degrees, it's gonna go

00:06:10.666 --> 00:06:12.558
all the way around and point
in the other direction,

00:06:12.558 --> 00:06:16.809
it's going to become, it's
going to map to ray OB.

00:06:16.809 --> 00:06:18.353
So I definitely believe that.

00:06:18.353 --> 00:06:20.689
OA is going to map to ray OB,

00:06:20.689 --> 00:06:24.762
and ray OC, if you rotate it 180 degrees,

00:06:24.762 --> 00:06:28.235
is going to map to ray OD.

00:06:28.235 --> 00:06:30.896
And so this first statement is true.

00:06:30.896 --> 00:06:34.162
If ray OA and ray OC are each rotated

00:06:34.162 --> 00:06:37.286
180 degrees about point
O, they must map to

00:06:37.286 --> 00:06:40.113
ray OB and OD respectively.

00:06:40.113 --> 00:06:41.374
And when people say respectively,

00:06:41.374 --> 00:06:42.610
they're saying in the same order.

00:06:42.610 --> 00:06:45.744
That ray OA maps to ray OB,

00:06:45.744 --> 00:06:49.849
and that ray OC maps to ray OD.

00:06:49.849 --> 00:06:52.409
And we saw that, ray OA maps,

00:06:52.409 --> 00:06:54.792
if you rotate it all the
way around 180 degrees,

00:06:54.792 --> 00:06:59.001
it'll map to OB, and then OC
if you rotate 180 degrees,

00:06:59.001 --> 00:07:02.247
will map to OD.

00:07:02.247 --> 00:07:05.639
So I'm feeling good about
that first sentence.

00:07:05.639 --> 00:07:08.728
If two rays are rotated
by the same amount,

00:07:08.728 --> 00:07:11.408
the angle between them will not change.

00:07:11.408 --> 00:07:14.191
Yeah, I could, yeah.

00:07:14.191 --> 00:07:17.399
Especially if they are rotated

00:07:17.399 --> 00:07:21.287
around, yeah, I'll go with that.

00:07:21.287 --> 00:07:22.983
If two rays are rotated
by the same amount,

00:07:22.983 --> 00:07:24.916
the angle between them will not change.

00:07:24.916 --> 00:07:29.581
So if we rotate both of
these rays by 180 degrees,

00:07:29.581 --> 00:07:33.698
then we've essentially
mapped to OB and OD.

00:07:33.698 --> 00:07:35.722
Or another way to think about this angle,

00:07:35.722 --> 00:07:40.241
angle AOC, is going to map to angle BOD.

00:07:40.241 --> 00:07:44.110
And so the measure of those
angles are going to be the same.

00:07:44.110 --> 00:07:47.258
So phi must be equal to theta.

00:07:47.258 --> 00:07:49.807
So I actually like this
second statement a lot.

00:07:49.807 --> 00:07:51.214
So let's see this last statement.

00:07:51.214 --> 00:07:56.018
Rotations preserve lengths and angles.

00:07:56.018 --> 00:08:00.158
AB is congruent to CD.

00:08:00.158 --> 00:08:02.615
Actually, we don't know whether segment

00:08:02.615 --> 00:08:06.480
AB is congruent to CD,
they never told us that.

00:08:06.480 --> 00:08:09.318
We don't know how far
apart these things are.

00:08:09.318 --> 00:08:12.447
So we know that phi is equal to theta.

00:08:12.447 --> 00:08:15.038
So this statement right
over here is just suspect.

00:08:15.038 --> 00:08:17.991
And so actually I don't like that one.

00:08:17.991 --> 00:08:20.081
So I'm gonna go with
the first one which is,

00:08:20.081 --> 00:08:22.863
it takes a little bit of
visualization going on,

00:08:22.863 --> 00:08:26.366
but if you took angle
AOC and you rotated it

00:08:26.366 --> 00:08:29.206
180 degrees which means
take the corresponding rays,

00:08:29.206 --> 00:08:32.214
or the rays that make it up,
and rotate them 180 degrees,

00:08:32.214 --> 00:08:34.973
you get to angle BOD.

00:08:34.973 --> 00:08:37.373
And the angle between those rays,

00:08:37.373 --> 00:08:39.272
or the measure of the angles
we were just talking about,

00:08:39.272 --> 00:08:40.541
shouldn't change.

00:08:40.541 --> 00:08:42.490
So I feel really good
about this second choice.


Ways to cut a cube

WEBVTT
Kind: captions
Language: en

00:00:00.393 --> 00:00:04.708
What I wanna do in this video
is explore the types of 2D shapes

00:00:04.708 --> 00:00:09.300
we can construct,
by taking planar slices of cubes.

00:00:09.663 --> 00:00:11.158
So what am I talking about?

00:00:11.365 --> 00:00:13.648
Let's say we wanna deconstruct a square:

00:00:13.834 --> 00:00:19.581
How can we slice a cube with a plane
to get the intersection of this cube,

00:00:19.581 --> 00:00:21.978
and that plane to be a square?

00:00:22.369 --> 00:00:25.300
We'll imagine that that plane
we are to cut, just like this,

00:00:25.300 --> 00:00:27.277
the square is maybe the most obvious one,

00:00:27.277 --> 00:00:31.632
so it cuts the top right over there,

00:00:31.632 --> 00:00:35.950
it cuts the side, right over here,

00:00:35.950 --> 00:00:39.747
it cuts the side, I guess on the back
of the glass cube, where you will see it,

00:00:39.747 --> 00:00:41.405
right over there, dotted line,

00:00:41.405 --> 00:00:46.915
and then it cuts this, right over here.
So you can imagine a plane that did this,

00:00:46.917 --> 00:00:50.369
and if I wanna draw the broader plane,
I can draw it like this.

00:00:50.534 --> 00:00:53.568
Let me see if I can do a decent,
an adequate job

00:00:53.568 --> 00:00:57.736
at drawing the actual,
as you see you'd say a part of the plane,

00:00:57.736 --> 00:01:03.433
that is cutting this cube.
It will look - it could look something...

00:01:03.433 --> 00:01:06.585
it could look something like this.
And I can even color in

00:01:06.585 --> 00:01:16.028
the part of the plane that you could
actually see it the cube were opaque,

00:01:16.028 --> 00:01:20.086
if you couldn't see through it.
But if you could see through it

00:01:20.086 --> 00:01:23.317
you would see this dotted line,
and the plane would look like that.

00:01:23.317 --> 00:01:27.010
So a square is
a pretty straightforward thing to get,

00:01:27.010 --> 00:01:29.594
if you're doing a planar slice of a cube.

00:01:29.594 --> 00:01:32.952
But what about a rectangle?
How can you get that?

00:01:32.952 --> 00:01:34.601
And at any point, I encourage you:

00:01:34.601 --> 00:01:37.120
Pause the video and try to think about it
on your own:

00:01:37.120 --> 00:01:39.595
How can you get these shapes
that I'm talking about?

00:01:39.934 --> 00:01:43.447
Well for a rectangle
you can actually cut like this:

00:01:43.800 --> 00:01:46.593
So, if you cut this side like this,

00:01:46.593 --> 00:01:49.703
and then cut that side like that,

00:01:49.703 --> 00:01:54.072
and then you cut this side like that
- I think you'd see where this is going,

00:01:54.072 --> 00:01:56.758
this side like that,

00:01:56.758 --> 00:02:00.045
and then you cut the bottom,
right over there,

00:02:00.045 --> 00:02:03.648
then the intersection of the plane
that you are cutting with;

00:02:03.648 --> 00:02:05.687
so, the intersection, let's see:

00:02:05.687 --> 00:02:08.761
This could be the plane
that I'm actually cutting with,

00:02:08.761 --> 00:02:11.840
so the intersection of the plane
that I'm cutting with,

00:02:11.840 --> 00:02:17.773
and my cube is going to be
a rectangle.

00:02:17.773 --> 00:02:22.621
So it might look like this, and once again
let me shade in the stuff,

00:02:22.621 --> 00:02:27.638
if you kind of view this,
if you imagine the plane

00:02:27.638 --> 00:02:32.249
just like one of those huge blades
that magicians use to saw people in half,

00:02:32.249 --> 00:02:34.211
or pretend like they are - they give us

00:02:34.211 --> 00:02:36.023
the illusion of sawing people in half,

00:02:36.023 --> 00:02:37.706
it might look something like this.

00:02:37.706 --> 00:02:40.415
Ok! So you'll go like:
"Ok, that's not so hard to digest,

00:02:40.415 --> 00:02:45.765
that if I intersect a plane with a cube,
I can get a square, I can get a rectangle.

00:02:46.135 --> 00:02:48.017
But what about triangles?"

00:02:48.129 --> 00:02:51.507
Well, once again pause the video
if you think you can figure it out,

00:02:51.867 --> 00:02:56.541
triangles are not so bad.
You can cut this side over here,

00:02:56.541 --> 00:03:00.938
this side right over here,

00:03:00.938 --> 00:03:04.223
and this side right over here,

00:03:04.223 --> 00:03:07.200
and then, this is it - of course
I can keep drawing the plane,

00:03:07.200 --> 00:03:08.673
but I think you get the idea -

00:03:08.673 --> 00:03:09.914
this would be a triangle.

00:03:09.914 --> 00:03:12.735
There's different types of triangles
that you can construct.

00:03:12.735 --> 00:03:15.741
You could construct
an equilateral triangle,

00:03:15.741 --> 00:03:22.340
so as long as this cut is the same length
as this cut right over here,

00:03:22.340 --> 00:03:24.661
is the same length as that upper length,

00:03:24.661 --> 00:03:27.556
or the length that intersects
on this space of the cube,

00:03:27.556 --> 00:03:29.467
that's gonna be an equilateral triangle.

00:03:29.467 --> 00:03:34.779
If you pushed this point up more,
actually I'd do that in a different color,

00:03:34.779 --> 00:03:40.064
you were going to have
an isosceles triangle.

00:03:40.064 --> 00:03:44.234
If you were to bring this point
really, really close, like here,

00:03:44.234 --> 00:03:50.004
you would approach having a right angle,
but it wouldn't be quite a right angle:

00:03:50.004 --> 00:03:53.767
you'd still have these angles
who'd still be less than 90º,

00:03:53.767 --> 00:03:56.681
you can approach 90º.
So you can't quite have

00:03:56.681 --> 00:04:00.461
an exactly a right angle,
and so since you can't get to 90º,

00:04:00.461 --> 00:04:04.031
sure enough you can't get near to 91º,
so actually you're not gonna be able to do

00:04:04.031 --> 00:04:05.637
an obtuse triangle either.

00:04:05.966 --> 00:04:08.630
But you can do an equilateral,
you can do an isosceles,

00:04:08.630 --> 00:04:10.016
you can do scalene triangles.

00:04:10.016 --> 00:04:13.406
I guess you could say you could do
the different types of acute triangles.

00:04:13.600 --> 00:04:16.230
But now let's do
some really interesting things:

00:04:16.429 --> 00:04:21.816
Can you get a pentagon
by slicing a cube with a plane?

00:04:21.816 --> 00:04:24.870
And I really want you to pause the video
and think about it here,

00:04:24.870 --> 00:04:27.166
because that's such a fun thing.
Think about it:

00:04:27.166 --> 00:04:31.421
How can you get a pentagon
by slicing a cube with a plane?

00:04:31.778 --> 00:04:33.659
All right, so here I go,

00:04:33.659 --> 00:04:36.911
this is how you can get a pentagon
by slicing a cube with a plane:

00:04:36.911 --> 00:04:41.600
Imagine slicing the top
- we'll do it a little bit different -

00:04:41.796 --> 00:04:49.733
so imagine slicing the top,
right over there, like this,

00:04:49.733 --> 00:04:54.751
Imagine slicing this backside,
like that,

00:04:54.751 --> 00:04:58.166
this back side that you can see,
quite like that,

00:04:58.166 --> 00:05:05.688
now you slice this side,
right over here, like this,

00:05:05.688 --> 00:05:09.633
and then you slice this side
right over here, like this.

00:05:09.799 --> 00:05:13.437
This could be,
and alike I'm gonna draw the plane

00:05:13.437 --> 00:05:16.403
- yet maybe it won't be so obvious
if I try to draw the plane -

00:05:16.403 --> 00:05:20.777
but you get the actual idea:
if I slice this, the right angle

00:05:20.777 --> 00:05:24.108
(not any right angle - 90º -
but 'the right angle' - the proper angle.

00:05:24.108 --> 00:05:27.452
Actually I shouldn't use 'right angle',
that would confuse everything.)

00:05:27.452 --> 00:05:29.703
If I slice it in the proper angle,

00:05:29.703 --> 00:05:32.415
that I'm doing
with the intersection of my plane

00:05:32.415 --> 00:05:37.796
then my cube is going to be
this pentagon, right over here.

00:05:38.040 --> 00:05:42.779
Now let's up the stakes something,
let's up the stakes even more!

00:05:42.779 --> 00:05:44.794
What about a hexagon?

00:05:44.794 --> 00:05:48.363
Can I slice a cube in a way,
with a 2D plane,

00:05:48.363 --> 00:05:52.406
to get the intersection of the plane
on the cube being a hexagon?

00:05:53.070 --> 00:05:55.981
As you could imagine,
I wouldn't have asked you that question

00:05:55.981 --> 00:05:59.173
unless I could.
So let's see if we can do it.

00:05:59.480 --> 00:06:06.749
So if we slice this, right over there,
if we slice this bottom piece,

00:06:06.749 --> 00:06:12.423
right over there,
and then you slice this back side,

00:06:12.423 --> 00:06:21.436
like that, and then you slice
the side that we can see right over there,

00:06:21.436 --> 00:06:23.844
(This side, I could have made it
much straighter)

00:06:23.844 --> 00:06:26.069
So hopefully you get the idea!

00:06:26.069 --> 00:06:30.367
I can slice this cube
so that I can actually get a hexagon.

00:06:30.767 --> 00:06:34.282
So, hopefully, this gives you
a better appreciation

00:06:34.282 --> 00:06:37.116
for what you could actually do
with a cube,

00:06:37.116 --> 00:06:41.613
especially if you're busy slicing it
with large planar planes

00:06:41.613 --> 00:06:45.753
- or large planar blades, in some way -

00:06:45.761 --> 00:06:49.412
There's actually more to a cube
that you might have imagined in the past.

00:06:49.412 --> 00:06:53.061
When we think about it,
there's six sides to a cube,

00:06:53.061 --> 00:06:55.042
and so it's six surfaces to a cube,

00:06:55.042 --> 00:06:58.160
so you can cut
as many as six of the surfaces

00:06:58.160 --> 00:06:59.848
when you intersect it with a plane,

00:06:59.848 --> 00:07:02.227
and every time you cut
into one of those surfaces

00:07:02.227 --> 00:07:04.842
it forms a side.
So here we're cutting into four sides,

00:07:04.842 --> 00:07:07.371
here we're cutting into four surfaces
of four sides,

00:07:07.371 --> 00:07:10.129
here we're cutting into three,
here we're cutting into five

00:07:10.129 --> 00:07:12.705
- we're not cutting into the bottom
of the cube, here -

00:07:12.705 --> 00:07:17.651
and here we're cutting into all six
of the sides, all six of the surfaces,

00:07:17.651 --> 00:07:20.700
of the faces of this cube.