Vector components from magnitude and direction

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:01.817
- So we have two examples here,

00:00:01.817 --> 00:00:04.065
where we're given the
magnitude of a vector,

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and it's direction, and the
direction is by giving us

00:00:07.053 --> 00:00:10.312
an angle that it forms
with the positive x-axis.

00:00:10.312 --> 00:00:11.845
What we need to do is go from having

00:00:11.845 --> 00:00:14.450
this magnitude and this
angle, this direction,

00:00:14.450 --> 00:00:17.770
to figuring out what
the x and y components

00:00:17.770 --> 00:00:20.580
of this vector actually are.

00:00:20.580 --> 00:00:22.495
So, like always, pause this video

00:00:22.495 --> 00:00:25.453
and see if you can work
through this on your own.

00:00:26.531 --> 00:00:28.166
Alright, now let's work
through this together,

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and it's really just gonna involve

00:00:29.340 --> 00:00:31.282
a little bit of trigonometry.

00:00:31.282 --> 00:00:34.040
So, we wanna break this vector
down into it's horizontal

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and it's vertical components,
or it's x and y components.

00:00:37.105 --> 00:00:38.842
So I could draw it's,

00:00:38.842 --> 00:00:41.932
I could draw it's horizontal component

00:00:41.932 --> 00:00:44.043
just like this.

00:00:44.043 --> 00:00:46.392
So it's gonna look something like--

00:00:46.392 --> 00:00:48.963
no, let me draw it, I can
do a better job than that.

00:00:48.963 --> 00:00:52.921
So, it's gonna look something like that.

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That's it's horizontal component,

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or it's x component.

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And then it's vertical component

00:00:58.387 --> 00:01:00.839
is going to look something like this.

00:01:00.839 --> 00:01:03.776
It's going to look like that.

00:01:03.776 --> 00:01:05.079
That's it's vertical
component, and notice,

00:01:05.079 --> 00:01:06.560
if you add the horizontal component

00:01:06.560 --> 00:01:07.991
and the vertical component,

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you are going to get your original vector.

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Now, I've just constructed
a right triangle,

00:01:12.792 --> 00:01:14.580
and in this type of a right triangle,

00:01:14.580 --> 00:01:16.547
I could just use, actually,
some of my most basic

00:01:16.547 --> 00:01:18.743
trig definitions, or the simplest form,

00:01:18.743 --> 00:01:20.557
which is the Soh-Cah-Toa definition

00:01:20.557 --> 00:01:21.609
of the trig functions.

00:01:21.609 --> 00:01:22.595
I might wanna break into the

00:01:22.595 --> 00:01:24.741
unit circle definition later on,

00:01:24.741 --> 00:01:27.285
but if I wanna figure out
the magnitude of this base,

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right over here, we see
that that is adjacent

00:01:29.558 --> 00:01:31.678
to this 50 degree angle.

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It's not the hypotenuse,

00:01:32.878 --> 00:01:34.743
it's the other side that forms the angle.

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And so what trig function deals

00:01:36.326 --> 00:01:38.073
with adjacent and hypotenuse?

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Well, we could just put
a little reminder here,

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Soh-Cah-Toah.

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Well, cosine deals with
adjacent over hypotenuse.

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So we could say that if I call,

00:01:51.386 --> 00:01:55.079
if I call this x the
length of our x component,

00:01:55.079 --> 00:01:57.658
we could say that the cosine,

00:01:57.658 --> 00:02:01.770
the cosine of 50 degrees of our angle

00:02:01.770 --> 00:02:05.244
is going to be equal to the
length of the adjacent side,

00:02:05.244 --> 00:02:09.219
x, over the length of our hypotenuse,

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which is the magnitude of the vector,

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over four.

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And so, if I wanna solve for x,

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I just multiply both sides by four.

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So I could get four times

00:02:20.329 --> 00:02:22.679
cosine of 50 degrees,

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four times cosine of 50 degrees,

00:02:25.131 --> 00:02:28.880
is equal to x.

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Now what about the y component?

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The y component, what is that going to be?

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Well, what, that side,

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the length of that side,
is opposite to the angle,

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the 50 degree angle.

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So what trig function deals
with opposite and hypotenuse?

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Well, that is sine.

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So we know that the sine of 50 degrees

00:02:51.107 --> 00:02:55.346
is going to be equal to y over
the length of the hypotenuse.

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Y over four, and so we can multiply

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both sides by four and we get,

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we get four times sine of 50 degrees,

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sin of 50 degrees,

00:03:07.527 --> 00:03:09.897
is equal to y.

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And so if I don't have a calculator,

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I could just write that this,

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this vector, I could write as,

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if I write it in the component form,

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it's x component if four
cosine of 50 degrees.

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And it's y component is
four sine of 50 degrees.

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And you might notice
something interesting here.

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I have cosine of the angle,

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the angle we formed with
the positive x axis,

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I have that for the x coordinate.

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I have sine on the y coordinate,

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and then I just multiply
it by the magnitude

00:03:42.345 --> 00:03:43.699
of the vector.

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Can I always do that?

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Well, it turns out you can,

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and this comes out of the
unit circle definition

00:03:47.760 --> 00:03:48.858
of the trig functions.

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The unit circle definition
of trig functions,

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cosine, if you have a unit circle,

00:03:53.276 --> 00:03:55.652
so unit circle looks like this.

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A unit circle has a radius of one.

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Cosine is the x coordinate of

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where you intersected the unit circle,

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and sine is the y coordinate.

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Or if you had a vector of magnitude one,

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it would be cosine of that angle,

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would be the x component,

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for the, if we had a unit
vector there in that direction.

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And then sine would be the y component.

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Well, we don't have a unit vector,

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our vector has a magnitude of four,

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it's four times bigger than a unit vector,

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so each of the components are going

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to be four times bigger.

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So that's why we multiply the x,

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the cosine of fifty by four
to get the x component,

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and we take the sine of 50,

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and we multiply it by four
to get the y component.

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And that's gonna come in handy

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when we think about this one over here.

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But we can get our calculator out,

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and approximate what
these are going to be.

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So let me get it out.

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Where's my calculator?

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Alright, there we go.

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So, 50, I'm in degree mode,

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gotta make sure, so 50 degrees.

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I'm gonna take the cosine,

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and I'm gonna multiply it times four.

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So times 4, is equal to two,

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approximately 2.57.

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So this is approximately equal to 2.57,

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is our x component, and
then our y component.

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Our y component, if I take 50 degrees,

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and if I take the sine of it,

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and then multiply it by four,

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I get approximately 3.06.

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3.06, and we see that over here,

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this x component looks
like it's a little more

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than two and a half,

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and this y component looks like

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it's slightly more than three.

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So it all worked out, even
though this is kind of

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a sloppy, hand drawn graph.

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Alright, now let's tackle this one.

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This is interesting cause this,

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the terminal point, when we
draw it in the standard form,

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is in the second quadrant.

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So what would be the x
and y components here,

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and you can immediately tell,

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because we're in the second quadrant,

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our x component is gonna be negative,

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and our y component is going to be,

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is going to be positive.

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Well, we could just resort exactly

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to what we did just there.

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We could say this vector

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is going to be the magnitude,

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it's x component is going
to be it's magnitude

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times the cosine of the angle that forms

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at the positive x axis.

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Cosine of 135 degrees.

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And it's y component is going to be

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the magnitude times the sine of the angle

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that forms with the positive x axis,

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and we'd be done.

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And we can evaluate each of these things,

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so we could get, let me get
my calculator out again.

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So if we take 135 degrees,

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and if I take the cosine of that,

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and then multiply it by ten,

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times ten, I get approximately

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negative 7.07.

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So that's approximately negative 7.07,

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and then if I take the sine,

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we're going to see something very similar.

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So 135 degrees, and I take the sine of it,

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I get positive .707,

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and so let me multiply
that times the magnitude,

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times ten,

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is equal to 7.07.

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So 7.07,

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and we see that over here.

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This is roughly, looks like
it's a little more than seven,

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looks like it's a little
bit more than seven

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in that direction.

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And we'd be done, and you might say well--

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once again, how did this work?

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Well, if I had a unit circle

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right over here.

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And if I had a unit vector,

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so it's terminal point would
sit on the unit circle,

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that went in the exact same direction,

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it still formed 135 degrees,

00:07:29.241 --> 00:07:31.310
this point right over here,

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it would have the coordinates

00:07:32.409 --> 00:07:34.247
cosine of 135 degrees,

00:07:34.247 --> 00:07:35.703
sine of 135 degrees,

00:07:35.703 --> 00:07:36.853
let me make it a little bit more visible,

00:07:36.853 --> 00:07:38.155
this point right over here.

00:07:38.155 --> 00:07:40.990
Cosine 135, sine of 135

00:07:40.990 --> 00:07:42.165
would be it's coordinates.

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Or for this unit vector right over here,

00:07:44.234 --> 00:07:45.690
that going in that direction,

00:07:45.690 --> 00:07:48.678
it's x component would be cosine of 135,

00:07:48.678 --> 00:07:51.309
and it's y component would be sine of 135.

00:07:51.309 --> 00:07:52.586
Well the vector that we care about

00:07:52.586 --> 00:07:54.169
has ten times the magnitude

00:07:54.169 --> 00:07:55.855
of a unit vector in that direction.

00:07:55.855 --> 00:07:57.285
So it's x component is going to have

00:07:57.285 --> 00:07:58.869
ten times the magnitude,

00:07:58.869 --> 00:08:00.887
and so is it's y component.

00:08:00.887 --> 00:08:02.394
And you could even resort to

00:08:02.394 --> 00:08:04.769
your Soh-Cah-Toa definitions
to figure this out.

00:08:04.769 --> 00:08:07.221
We could construct a
right triangle if we like,

00:08:07.221 --> 00:08:11.997
we could say here is,

00:08:11.997 --> 00:08:14.210
we could say here is my x component,

00:08:15.620 --> 00:08:16.846
that is my x component,

00:08:16.846 --> 00:08:18.276
it's gonna be something like that,

00:08:18.276 --> 00:08:19.221
and then there--

00:08:19.221 --> 00:08:21.954
and then my y component is going to look

00:08:21.954 --> 00:08:25.530
something like, I'm drawing
it a little bit imprecise,

00:08:25.530 --> 00:08:28.534
but I could draw that, let me
draw that a little bit better.

00:08:28.534 --> 00:08:31.565
My y component is gonna
look something like that,

00:08:31.565 --> 00:08:36.024
right triangle, and so
if I want this magnitude,

00:08:36.024 --> 00:08:38.348
the magnitude of this bottom side,

00:08:38.348 --> 00:08:39.523
I would say well, look,

00:08:39.783 --> 00:08:44.708
I know if this is 135 degrees,

00:08:44.708 --> 00:08:46.700
this angle right over here
is supplementary to that,

00:08:46.700 --> 00:08:49.712
so 45 degree angle,

00:08:49.712 --> 00:08:51.040
and so what trig function

00:08:51.040 --> 00:08:52.751
deals with adjacent and hypotenuse,

00:08:52.751 --> 00:08:54.105
Well cosine.

00:08:54.105 --> 00:08:57.579
So I could say cosine of 45 degrees

00:08:57.579 --> 00:08:59.801
is equal to the length of this,

00:08:59.801 --> 00:09:01.921
so let me just call the length of that,

00:09:01.921 --> 00:09:05.011
oh I don't know, let me just call it x,

00:09:05.011 --> 00:09:06.799
so the length of it, x,

00:09:06.799 --> 00:09:09.404
over the hypotenuse, over ten.

00:09:09.404 --> 00:09:10.809
Multiply both sides by ten,

00:09:10.809 --> 00:09:14.104
you get ten times cosine of 45 degrees

00:09:14.104 --> 00:09:16.198
is equal to x.

00:09:16.198 --> 00:09:18.088
And cosine of 45 degrees is

00:09:18.088 --> 00:09:19.442
square root of two over two,

00:09:19.442 --> 00:09:22.957
so this is going to be
five square roots of two

00:09:22.957 --> 00:09:25.664
is equal to x, and you might say,

00:09:25.664 --> 00:09:27.963
wait, wait, I thought this
was supposed to be negative.

00:09:27.963 --> 00:09:29.470
Well, the way we just solved it right now,

00:09:29.470 --> 00:09:30.364
with our right triangle,

00:09:30.364 --> 00:09:32.509
we just figured out our
magnitude of this side.

00:09:32.509 --> 00:09:33.658
And then we would just have to reason,

00:09:33.658 --> 00:09:36.621
ok, we're not going five square
roots of two to the right,

00:09:36.621 --> 00:09:38.766
we're going five square
roots of two to the left.

00:09:38.766 --> 00:09:41.908
So, the magnitude in the x direction,

00:09:41.908 --> 00:09:44.666
or not the magnitude, our x component,

00:09:44.666 --> 00:09:46.301
I should say, is going to be negative

00:09:46.301 --> 00:09:49.162
five square roots of two.

00:09:49.162 --> 00:09:51.001
And you could do the exact
same reasoning to say,

00:09:51.001 --> 00:09:52.303
well ten cosine of f--

00:09:52.303 --> 00:09:54.474
it's ten sine of 45 degrees

00:09:54.474 --> 00:09:58.245
is going to give us the
magnitude of the y component,

00:09:58.245 --> 00:10:00.492
and that's actually going
to be our y component.

00:10:00.492 --> 00:10:02.152
So, same exact thing,

00:10:02.152 --> 00:10:06.689
this is going to be
ten sine of 45 degrees,

00:10:06.689 --> 00:10:09.346
which is five square roots of two,

00:10:09.346 --> 00:10:10.521
and if we took our calculator out,

00:10:10.521 --> 00:10:11.849
they would evaluate to approximately

00:10:11.849 --> 00:10:14.372
these things right over there.


Angles of vectors from components

WEBVTT
Kind: captions
Language: en

00:00:00.369 --> 00:00:01.446
- [Voiceover] So what we're
going to do in this video

00:00:01.446 --> 00:00:04.109
is look at a series of
vectors, and we're gonna

00:00:04.109 --> 00:00:06.389
draw them in standard form,
where their initial points

00:00:06.389 --> 00:00:08.932
or their tails are
gonna sit at the origin.

00:00:08.932 --> 00:00:10.106
And we're gonna see if we can figure out

00:00:10.106 --> 00:00:12.647
the angles that they
form, the positive angles

00:00:12.647 --> 00:00:15.321
that they form with the positive x-axis.

00:00:15.321 --> 00:00:16.724
And like always, pause this video and see

00:00:16.724 --> 00:00:18.159
if you could figure out
what these thetas are

00:00:18.159 --> 00:00:19.461
going to be on your own, and we're gonna

00:00:19.461 --> 00:00:21.398
figure 'em out in degrees.

00:00:21.398 --> 00:00:24.200
So in this first one, I have the vector u.

00:00:24.200 --> 00:00:26.343
If we were to write it
in unit-vector notation,

00:00:26.343 --> 00:00:28.363
sometimes this is called
engineering notation,

00:00:28.363 --> 00:00:30.365
we would say it's three
times the unit vector

00:00:30.365 --> 00:00:33.719
in the horizontal direction,
the i unit vector,

00:00:33.719 --> 00:00:36.606
plus four times the unit vector
in the vertical direction

00:00:36.606 --> 00:00:39.288
or you could view this as
the x-component is three

00:00:39.288 --> 00:00:41.426
and the y-component is
four, and you see that here.

00:00:41.426 --> 00:00:44.908
If you start at the origin,
we're gonna move three

00:00:44.908 --> 00:00:46.766
in the horizontal direction,
and we're gonna move

00:00:46.766 --> 00:00:49.365
four in the vertical direction.

00:00:49.365 --> 00:00:51.705
Now to figure out this theta, there's

00:00:51.705 --> 00:00:52.826
a couple of ways to think about it.

00:00:52.826 --> 00:00:54.808
We could just construct a right triangle

00:00:54.808 --> 00:00:56.182
for this one in particular.

00:00:56.182 --> 00:00:59.006
We could say, okay, the
x-coordinate is three.

00:00:59.006 --> 00:01:02.587
So if we were to create
a right triangle here,

00:01:02.587 --> 00:01:05.028
this side would have a length of three,

00:01:05.028 --> 00:01:08.911
and then we have a height of four

00:01:08.911 --> 00:01:10.606
or the y-coordinate is four.

00:01:10.606 --> 00:01:14.608
So this side has a length of four,

00:01:14.608 --> 00:01:17.124
and we know just from
even our basic SOHCAHTOA

00:01:17.124 --> 00:01:19.907
definitions of trig functions
that what trig function

00:01:19.907 --> 00:01:22.367
involves the opposite of an angle?

00:01:22.367 --> 00:01:25.164
So the opposite of an angle,
and the adjacent of an angle?

00:01:25.164 --> 00:01:26.269
Well, tangent does.

00:01:26.269 --> 00:01:27.450
SOHCAHTOA.

00:01:27.450 --> 00:01:32.070
So we know, we know that
the tangent of theta

00:01:32.070 --> 00:01:35.030
is going to be equal to the
length of the opposite side,

00:01:35.030 --> 00:01:37.887
which is four, over the length of

00:01:37.887 --> 00:01:40.967
the adjacent side, over three.

00:01:40.967 --> 00:01:42.949
And so if we wanted to solve for theta,

00:01:42.949 --> 00:01:45.889
we could just say that theta is equal to

00:01:45.889 --> 00:01:47.748
the inverse tangent.

00:01:47.748 --> 00:01:49.409
Sometimes people say arc tangent

00:01:49.409 --> 00:01:53.424
of 4/3, of four over ...

00:01:55.024 --> 00:01:58.292
four over three, and let's evaluate this.

00:01:58.292 --> 00:02:01.654
And I'm gonna get my
calculator out to do it.

00:02:01.654 --> 00:02:06.654
So I wanna take four divided by three,

00:02:06.932 --> 00:02:09.394
which is that, and I
had already had pressed

00:02:09.394 --> 00:02:11.152
this button right over here that takes my

00:02:11.152 --> 00:02:14.054
tangent and makes it
into an inverse tangent.

00:02:14.054 --> 00:02:16.994
So I'm gonna take the
inverse tangent of this,

00:02:16.994 --> 00:02:21.994
and I get, I get roughly 53.1 degrees.

00:02:22.256 --> 00:02:25.568
So this is approximately

00:02:25.568 --> 00:02:29.613
53.1 degrees, which looks about right.

00:02:29.613 --> 00:02:30.992
This looks like a little
bit more, even though

00:02:30.992 --> 00:02:33.288
I didn't draw it
completely super precisely.

00:02:33.288 --> 00:02:34.211
I hand-drew this.

00:02:34.211 --> 00:02:36.608
It looks a little bit more
than a 45 degree angle,

00:02:36.608 --> 00:02:39.650
so that feels, that feels pretty good.

00:02:39.650 --> 00:02:41.149
Now another thing you might have said,

00:02:41.149 --> 00:02:42.005
"Hey. Well, look.

00:02:42.005 --> 00:02:44.043
"Four is the y-component.

00:02:44.043 --> 00:02:47.163
"Three is the x-component,
and so maybe tangent

00:02:47.163 --> 00:02:49.048
"of theta is always going to be

00:02:49.048 --> 00:02:52.342
"the y-component over the x-component."

00:02:52.342 --> 00:02:54.205
And that, in fact, is the case.

00:02:54.205 --> 00:02:55.403
And that actually comes
straight out of the

00:02:55.403 --> 00:02:57.987
unit circle definition
of the trig functions,

00:02:57.987 --> 00:02:59.764
where you could say that if you have

00:02:59.764 --> 00:03:03.133
a unit circle, if you have, I
could draw it right over here.

00:03:03.133 --> 00:03:06.112
So let me draw the coordinates.

00:03:06.112 --> 00:03:08.749
So if I were to draw a unit
circle right over here,

00:03:08.749 --> 00:03:10.629
and if I were to have some line,

00:03:10.629 --> 00:03:12.045
here we're thinking about vectors,

00:03:12.045 --> 00:03:14.791
that the angle formed
with the positive x-axis,

00:03:14.791 --> 00:03:18.488
the tangent of that
angle, tangent of theta

00:03:18.488 --> 00:03:20.250
is going to be the y-coordinate, where we

00:03:20.250 --> 00:03:22.710
intersect the circle,
over the x-coordinate.

00:03:22.710 --> 00:03:23.970
And so you could imagine if you made

00:03:23.970 --> 00:03:26.444
a unit circle right here,
if you made a unit circle

00:03:26.444 --> 00:03:30.168
right over here, the ratio
between the y-coordinate

00:03:30.168 --> 00:03:31.930
and the x-coordinate of
this point right over here

00:03:31.930 --> 00:03:33.627
where we intersect the
circle is going to be

00:03:33.627 --> 00:03:35.865
the same thing as the ratio
between four and three.

00:03:35.865 --> 00:03:37.725
So this is going to be,
this also is going to be

00:03:37.725 --> 00:03:40.826
one and 1/3 or 4/3.

00:03:40.826 --> 00:03:42.965
So either way, you could
think of, when you think of

00:03:42.965 --> 00:03:45.569
vectors, the tangent of
the angle that it forms

00:03:45.569 --> 00:03:49.030
with the positive x-axis is going to be,

00:03:49.030 --> 00:03:53.350
is going to be the y-component
over the x-component.

00:03:53.350 --> 00:03:55.990
So it's nice when everything
kinda fits together like that.

00:03:55.990 --> 00:03:57.245
So let's leverage that to figure out

00:03:57.245 --> 00:03:59.868
this, to figure out this
angle right over there.

00:03:59.868 --> 00:04:04.868
Well, we could say that tangent of theta

00:04:05.109 --> 00:04:09.509
is going to be equal to the y-component.

00:04:09.509 --> 00:04:12.250
The y-component over the x-component.

00:04:12.250 --> 00:04:15.308
Over, do this in a new color

00:04:15.308 --> 00:04:18.008
or let's see, over negative five.

00:04:18.008 --> 00:04:20.879
So it's gonna be negative 6/5.

00:04:20.879 --> 00:04:23.775
Or we could say theta.

00:04:23.775 --> 00:04:25.294
So let's be a little bit careful.

00:04:25.294 --> 00:04:27.651
So we could say, we could say that theta

00:04:27.651 --> 00:04:32.651
is equal to, is equal
to the inverse tangent.

00:04:32.753 --> 00:04:35.330
Let me do that same color.

00:04:35.330 --> 00:04:40.012
Inverse tangent of, I'll
just write it like this,

00:04:40.012 --> 00:04:43.530
negative 6/5, but I'm going
to put a little question mark

00:04:43.530 --> 00:04:46.452
here to see if we feel
good about the answer

00:04:46.452 --> 00:04:47.812
that you get when we do this.

00:04:47.812 --> 00:04:48.848
So let's do that.

00:04:48.848 --> 00:04:53.848
So if we do, if we do six divided by five,

00:04:54.053 --> 00:04:55.294
which'll equal to that, and then we

00:04:55.294 --> 00:04:56.134
want to make this negative.

00:04:56.134 --> 00:04:57.869
So that's negative 6/5, and we're gonna

00:04:57.869 --> 00:04:58.953
take the inverse tangents.

00:04:58.953 --> 00:04:59.994
I already pressed this button, so this

00:04:59.994 --> 00:05:01.894
is going to be an inverse
tangent, not tangent.

00:05:01.894 --> 00:05:06.894
I get negative, roughly
negative 50.2 degrees.

00:05:07.831 --> 00:05:12.831
So this is approximately
negative 50.2 degrees.

00:05:12.854 --> 00:05:14.134
Well, does that look right?

00:05:14.134 --> 00:05:14.675
Well, no.

00:05:14.675 --> 00:05:17.173
Theta looks like it's over 90 degrees.

00:05:17.173 --> 00:05:21.134
Negative 50.2 degrees,
negative 50.2 degrees

00:05:21.134 --> 00:05:25.617
is actually giving us this
angle right over here.

00:05:25.617 --> 00:05:27.291
So that's giving us, that's specifying

00:05:27.291 --> 00:05:29.314
another vector, if you think about a line,

00:05:29.314 --> 00:05:33.104
another line that would
have the same tangent.

00:05:33.104 --> 00:05:33.788
The same tangent.

00:05:33.788 --> 00:05:35.211
So it's really important to visualize this

00:05:35.211 --> 00:05:36.268
and think about this,
and the reason why it is

00:05:36.268 --> 00:05:39.533
is because the arc tangent or
the inverse tangent functions

00:05:39.533 --> 00:05:43.353
in calculators, they
will give you an angle

00:05:43.353 --> 00:05:45.154
that is between negative 90 degrees and

00:05:45.154 --> 00:05:47.450
positive 90 degrees, so something

00:05:47.450 --> 00:05:49.269
that's in the fourth or first quadrant.

00:05:49.269 --> 00:05:51.571
While here, we have something
that's in the second quadrant.

00:05:51.571 --> 00:05:53.608
So we have to make sure that
we're thinking about it right

00:05:53.608 --> 00:05:56.130
so that we could make the
appropriate adjustment.

00:05:56.130 --> 00:05:58.993
So that would give the
case if we were looking

00:05:58.993 --> 00:06:02.651
at a line like that or a vector like that.

00:06:02.651 --> 00:06:05.761
So in order to figure out
what, in order to figure out

00:06:05.761 --> 00:06:08.118
the actual angle, what we want to do

00:06:08.118 --> 00:06:10.911
is add 180 degrees to it.

00:06:10.911 --> 00:06:15.034
So to get the vector that
goes in the other direction.

00:06:15.034 --> 00:06:19.151
So you want to add, so theta
is going to be equal to

00:06:19.151 --> 00:06:21.804
or I could say approximately equal to

00:06:21.804 --> 00:06:26.804
negative 50.2 degrees plus 180 degrees.

00:06:27.131 --> 00:06:28.997
So let's do that.

00:06:28.997 --> 00:06:33.997
So let me plus 180 is equal to

00:06:34.975 --> 00:06:39.136
approximately 129.8 degrees.

00:06:39.136 --> 00:06:44.136
So theta is approximately
equal to 129.8 degrees.

00:06:44.934 --> 00:06:45.812
Did I get that right?

00:06:45.812 --> 00:06:46.977
I have a very short memory.

00:06:46.977 --> 00:06:47.897
Okay, yeah; that's right.

00:06:47.897 --> 00:06:51.918
129.8 degrees, and that looks much better.

00:06:51.918 --> 00:06:55.083
That looks clearly is
an angle we're gonna go.

00:06:55.083 --> 00:06:56.052
That take us to 90 degrees,

00:06:56.052 --> 00:06:58.015
and then we're going above 90 degrees.

00:06:58.015 --> 00:07:00.110
Now another way you could
have thought about this

00:07:00.110 --> 00:07:04.853
is what you know from SOHCATOA
and just right triangles.

00:07:04.853 --> 00:07:07.935
We could construct a right triangle where,

00:07:07.935 --> 00:07:11.054
we construct a right
triangle using some colors.

00:07:11.054 --> 00:07:14.212
Well, we know, so I could
maybe just draw it like this.

00:07:14.212 --> 00:07:18.295
So this is the height of
it, and what is that height?

00:07:18.295 --> 00:07:21.869
Well, the height is going to be six,

00:07:21.869 --> 00:07:26.869
and then the base, the
base right over here,

00:07:27.011 --> 00:07:30.254
what is that length going to be?

00:07:30.254 --> 00:07:32.149
Well, we know that we're
going from the origin,

00:07:32.149 --> 00:07:33.795
we're going five back, but
if we think about just the

00:07:33.795 --> 00:07:35.713
absolute value, the length of that line,

00:07:35.713 --> 00:07:38.253
that is going to be five, and so we could

00:07:38.253 --> 00:07:41.652
figure out just using
right triangles this angle.

00:07:41.652 --> 00:07:44.269
Maybe we call that x, and so we could say

00:07:44.269 --> 00:07:47.753
that the tangent of x, tangent of x

00:07:47.753 --> 00:07:52.753
is equal to six over five,
opposite over adjacent.

00:07:52.972 --> 00:07:55.943
Six over five.

00:07:56.623 --> 00:07:59.829
Or that x, or you could say

00:07:59.829 --> 00:08:04.046
x is equal to the inverse tangent

00:08:04.046 --> 00:08:07.688
of 6/5, and what is that going to be?

00:08:07.688 --> 00:08:12.688
So, if we take six divided by five, 1.2.

00:08:13.728 --> 00:08:17.004
And then we're going to take
the inverse tangent of that,

00:08:17.004 --> 00:08:21.944
and we get approximately 50.2 degrees.

00:08:21.944 --> 00:08:26.944
So x is approximately 50.2 degrees,

00:08:27.863 --> 00:08:29.064
which looks right.

00:08:29.064 --> 00:08:30.003
But remember, we're not
trying to figure out x,

00:08:30.003 --> 00:08:31.324
we're trying to figure out,

00:08:31.324 --> 00:08:34.648
we are trying to figure out what theta,

00:08:34.648 --> 00:08:37.726
what theta is, and you
can see that x and theta

00:08:37.726 --> 00:08:39.107
are supplementary.

00:08:39.107 --> 00:08:41.229
So theta plus x is going to be 180 or

00:08:41.229 --> 00:08:44.146
theta's gonna be 180 minus that.

00:08:44.146 --> 00:08:45.985
So 180 minus that.

00:08:45.985 --> 00:08:47.263
So let's just put a negative
sign in front of that

00:08:47.263 --> 00:08:49.968
and add 180, and that might look familiar.

00:08:49.968 --> 00:08:54.968
So that gets to us to
129, roughly 129.8 degrees

00:08:55.469 --> 00:08:59.037
is what exactly what we got before.


More examples finding vector angles

WEBVTT
Kind: captions
Language: en

00:00:00.139 --> 00:00:01.741
- [Voiceover] Let's get some more practice

00:00:01.741 --> 00:00:05.618
finding the angle, in these
cases the positive angle,

00:00:05.618 --> 00:00:08.625
between the positive X axis and a vector

00:00:08.625 --> 00:00:10.900
drawn in standard form
where it's initial point,

00:00:10.900 --> 00:00:14.360
or it's tail, is sitting at the origin.

00:00:14.360 --> 00:00:18.191
Here for vector A we can write
it in two different ways.

00:00:18.191 --> 00:00:19.216
In engineering notation

00:00:19.216 --> 00:00:21.852
it would be -2 times a unit vector I,

00:00:21.852 --> 00:00:23.977
that's the unit vector in the X direction,

00:00:23.977 --> 00:00:26.670
minus four times the unit
vector in the Y direction,

00:00:26.670 --> 00:00:28.222
or we could just say
it's X component is -2,

00:00:28.222 --> 00:00:29.742
it's Y component is -4.

00:00:29.742 --> 00:00:30.821
And we see that here.

00:00:30.821 --> 00:00:34.688
If we're starting at the
origin we go two to the left

00:00:34.688 --> 00:00:38.310
and we go four down to
get to the terminal point

00:00:38.310 --> 00:00:40.829
or the head of the vector.

00:00:40.829 --> 00:00:42.582
Also figure out what theta is.

00:00:42.582 --> 00:00:45.315
What we've seen before when
we're thinking about vectors

00:00:45.315 --> 00:00:48.136
drawn in standard form,
we could say the tangent

00:00:48.136 --> 00:00:51.712
of this angle is going to
be equal to the Y component

00:00:51.712 --> 00:00:53.813
over the X component.

00:00:53.813 --> 00:00:56.224
So the Y component is -4

00:00:56.224 --> 00:01:00.334
and the X component is -2.

00:01:00.334 --> 00:01:02.714
And in the previous video
we explained why this is,

00:01:02.714 --> 00:01:05.337
it really comes straight out
of the unit circle definition

00:01:05.337 --> 00:01:07.578
of trig functions, tangent of theta

00:01:07.578 --> 00:01:10.736
is equal to the Y coordinate
over the X coordinate

00:01:10.736 --> 00:01:14.462
of where a line that
defines an angle intersects

00:01:14.462 --> 00:01:16.018
the unit circle.

00:01:16.018 --> 00:01:17.678
And I encourage you to watch that video

00:01:17.678 --> 00:01:19.976
if that doesn't make much sense.

00:01:19.976 --> 00:01:24.574
But so we could say tangent
of theta is equal to two.

00:01:24.574 --> 00:01:27.557
And so we might want to say,
if we want to solve for theta,

00:01:27.557 --> 00:01:28.892
we could say theta is equal to

00:01:28.892 --> 00:01:33.141
the inverse tangent function of two.

00:01:33.141 --> 00:01:34.302
And I'm gonna put a question mark,

00:01:34.302 --> 00:01:36.554
and I think you might know why

00:01:36.554 --> 00:01:37.599
I'm putting that question mark.

00:01:37.599 --> 00:01:39.630
So let's see what that gets us.

00:01:39.630 --> 00:01:41.558
So if we were to take two,

00:01:41.558 --> 00:01:43.531
and I wanna take the inverse
tangent not just the tangent.

00:01:43.531 --> 00:01:46.793
I wanna figure out what angle
gives me a tangent of two.

00:01:46.793 --> 00:01:50.024
So inverse tangent,

00:01:50.024 --> 00:01:53.263
it's about 63.4 degrees roughly.

00:01:53.263 --> 00:01:55.903
So this gives me theta is approximately

00:01:55.903 --> 00:02:00.442
63.4 degrees.

00:02:02.118 --> 00:02:04.277
And I'm gonna put a question mark here.

00:02:04.277 --> 00:02:06.482
And I think you might sense why that is.

00:02:06.482 --> 00:02:08.061
If you don't, pause the video

00:02:08.061 --> 00:02:09.965
and think about why am I
putting a question mark here?

00:02:09.965 --> 00:02:13.738
Why does this angle look fishy?

00:02:13.738 --> 00:02:17.256
Well, it looks fishy because
an angle of 63.4 degrees

00:02:17.256 --> 00:02:20.403
would put us squarely

00:02:20.403 --> 00:02:22.423
in the first quadrant.

00:02:22.423 --> 00:02:27.124
If our vector looked like this,

00:02:27.124 --> 00:02:28.276
let me see if I can draw it.

00:02:32.321 --> 00:02:36.074
If our vector looked like this,

00:02:38.100 --> 00:02:41.547
so if our vector's
components were positive two

00:02:41.547 --> 00:02:46.366
and positive four then that
looks like a 63-degree angle.

00:02:46.366 --> 00:02:49.337
This looks like a 63-degree angle.

00:02:49.337 --> 00:02:50.707
But we're not in the first quadrant.

00:02:50.707 --> 00:02:53.958
Our vector A that we care
about is in the third quadrant.

00:02:53.958 --> 00:02:58.577
So it's clear that it's in
the exact opposite direction,

00:02:58.577 --> 00:03:00.237
and I think you see why.

00:03:00.237 --> 00:03:03.534
When we take the inverse tangent
function on our calculator

00:03:03.534 --> 00:03:06.764
it assumes that the angle
is between -90 degrees

00:03:06.764 --> 00:03:08.331
and positive 90 degrees.

00:03:08.331 --> 00:03:11.650
Well, here we have an angle
that's over 180 degrees.

00:03:11.650 --> 00:03:14.797
It's between 180 and 270 degrees.

00:03:14.797 --> 00:03:15.738
And so to find this angle,

00:03:15.738 --> 00:03:18.138
and this is why if you're
ever using the inverse tangent

00:03:18.138 --> 00:03:20.599
function on your calculator
it's very, very important,

00:03:20.599 --> 00:03:22.340
whether you're doing
vectors or anything else,

00:03:22.340 --> 00:03:26.078
to think about where does
your angle actually sit?

00:03:26.078 --> 00:03:28.202
What quadrant does it actually put you in

00:03:28.202 --> 00:03:30.456
because you might have
to adjust those figures.

00:03:30.456 --> 00:03:34.065
So, it's not going to be 63.4
degrees it's going to be that

00:03:34.065 --> 00:03:38.025
plus another 180 degrees to
go all the way over here.

00:03:38.025 --> 00:03:42.262
So, theta is going to be 180,
and I should say approximately

00:03:42.262 --> 00:03:47.262
'cause I still rounded,
180 plus 63.4 degrees

00:03:47.430 --> 00:03:50.992
is going to be 200 and, what is that?

00:03:50.992 --> 00:03:53.097
180 plus 60 is 240,

00:03:53.097 --> 00:03:57.636
so 243.4 degrees.

00:03:57.636 --> 00:03:58.959
So let's do one more.

00:03:58.959 --> 00:04:01.281
So here I have a vector
sitting in the fourth quadrant

00:04:01.281 --> 00:04:02.280
like we just did.

00:04:02.280 --> 00:04:03.661
Pause the video and see
if you can figure out

00:04:03.661 --> 00:04:07.956
the positive angle that it
forms with the positive X axis.

00:04:07.956 --> 00:04:10.423
Well, we could do the same drill

00:04:10.423 --> 00:04:12.500
and maybe we could skip a few steps here

00:04:12.500 --> 00:04:14.694
now that we've done it many times.

00:04:14.694 --> 00:04:18.700
We might wanna say that
the inverse tangent of,

00:04:18.700 --> 00:04:21.218
let me write it this way,
we might want to write,

00:04:21.218 --> 00:04:22.576
I'll do the same color.

00:04:22.576 --> 00:04:26.198
We might wanna say that
theta is equal to the inverse

00:04:26.198 --> 00:04:30.540
tangent of my Y component
over my X component

00:04:30.540 --> 00:04:34.423
of -6 over four, and we know what that is

00:04:34.423 --> 00:04:38.137
but let me just actually
not skip too many steps.

00:04:38.137 --> 00:04:42.406
So the inverse tangent of -1.5.

00:04:42.406 --> 00:04:46.560
- 1, -1.5 and once again,

00:04:46.560 --> 00:04:49.138
I get to get my calculator out

00:04:49.138 --> 00:04:53.685
and so 1.5 negative,

00:04:53.685 --> 00:04:55.926
and I wanna find the
inverse tangent of it,

00:04:55.926 --> 00:05:00.082
I get roughly -56.3 degrees.

00:05:00.082 --> 00:05:03.260
So this is approximately equal to

00:05:03.260 --> 00:05:06.754
- 53.6 degrees.

00:05:06.754 --> 00:05:09.647
Did I do that right, 56.3,

00:05:09.647 --> 00:05:13.745
56.3 degrees.

00:05:13.745 --> 00:05:16.426
And once again, I'm gonna
put the question marks here.

00:05:16.426 --> 00:05:17.483
And why did I do that?

00:05:17.483 --> 00:05:19.445
Because the angle that it's giving,

00:05:19.445 --> 00:05:21.406
and this isn't wrong
actually in this case,

00:05:21.406 --> 00:05:23.369
it's just not giving
us the positive angle.

00:05:23.369 --> 00:05:25.470
Because if you start the positive X axis

00:05:25.470 --> 00:05:28.186
and you were to go clockwise,

00:05:28.186 --> 00:05:30.565
well now your angle is
going to be negative,

00:05:30.565 --> 00:05:33.190
and that is -56.3 degrees.

00:05:33.190 --> 00:05:36.881
But we wanna figure out the
positive angle right over here.

00:05:36.881 --> 00:05:38.286
So, there's a couple of ways

00:05:38.286 --> 00:05:40.527
that you could think about doing it.

00:05:40.527 --> 00:05:41.804
One way to think about it is

00:05:41.804 --> 00:05:44.137
well to go from this negative angle

00:05:44.137 --> 00:05:45.844
to the positive version of it

00:05:45.844 --> 00:05:48.142
we have to go completely around once.

00:05:48.142 --> 00:05:51.577
So we have to add 360 degrees.

00:05:51.577 --> 00:05:53.028
So if it's really

00:05:53.028 --> 00:05:56.963
approximately -56.3 degrees

00:05:56.963 --> 00:06:00.380
plus 360 degrees,

00:06:00.380 --> 00:06:03.259
which is going to be, what is that?

00:06:03.259 --> 00:06:04.749
So it's going to be,

00:06:04.749 --> 00:06:07.745
so it's going to be approximately,

00:06:07.745 --> 00:06:11.633
see if I subtracted 50 degrees
I would get to 310 degrees,

00:06:11.633 --> 00:06:15.824
I subtract another six
degrees, so it's 304 degrees,

00:06:15.824 --> 00:06:17.219
and then .3,

00:06:17.219 --> 00:06:21.478
so 303.7 degrees.

00:06:21.478 --> 00:06:22.558
Did I do that right?

00:06:22.558 --> 00:06:25.507
Let's see, if I add this .3 to the seven,

00:06:25.507 --> 00:06:28.579
that's gonna get to
304, then at 310 to 360.

00:06:28.579 --> 00:06:29.566
So there ya go.

00:06:29.566 --> 00:06:33.304
That is our positive angle that we form.

00:06:33.304 --> 00:06:36.937
So always really think about
what they're asking from you,

00:06:36.937 --> 00:06:38.435
or what a question is asking from you.

00:06:38.435 --> 00:06:40.554
Anyway, you get the idea.


Combined vector operations example

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.524
- [Voiceover] So what we have
right over here, we have the

00:00:02.524 --> 00:00:06.482
vector U and we've defined it
by giving its X & Y components

00:00:06.482 --> 00:00:10.522
it's a two-dimensional vector,
and we have the vector W.

00:00:10.522 --> 00:00:12.665
And we've graphed them, the vector U,

00:00:12.665 --> 00:00:16.565
its X component is two, its
Y component is negative one.

00:00:16.565 --> 00:00:19.839
So if we put its initial point
at the origin, the terminal

00:00:19.839 --> 00:00:24.285
point, or its head, will
be at the point two, comma,

00:00:24.285 --> 00:00:26.764
negative one, which is right over there.

00:00:26.764 --> 00:00:29.980
And for vector W, it's
negative five, comma, five.

00:00:29.980 --> 00:00:33.033
So its X component is negative five.

00:00:33.033 --> 00:00:36.121
So if we start at the origin
we would move five to the left.

00:00:36.121 --> 00:00:38.501
And its Y component is
positive five, so we

00:00:38.501 --> 00:00:42.436
would move five up then to
get to the head of the vector,

00:00:42.436 --> 00:00:45.037
or to get to the terminal point.

00:00:45.037 --> 00:00:48.241
Now given these two vectors,
what we wanna do is evaluate

00:00:48.241 --> 00:00:52.420
what three times the
vector U, plus one fifth

00:00:52.420 --> 00:00:54.974
times the vector W is, and I encourage you

00:00:54.974 --> 00:00:57.911
to pause the video and to give a go at it.

00:00:57.911 --> 00:01:00.070
Well three and one fifth are scalers.

00:01:00.070 --> 00:01:02.079
They are going to scale these
vectors and we're gonna see

00:01:02.079 --> 00:01:04.493
them, see that happen visually.

00:01:04.493 --> 00:01:07.373
So we're gonna scale up
vector U by three, we're gonna

00:01:07.373 --> 00:01:11.215
scale down vector W, we're
gonna multiply it by one fifth,

00:01:11.215 --> 00:01:13.595
and then we're gonna add
the resulting vectors.

00:01:13.595 --> 00:01:14.814
So let's do that.

00:01:14.814 --> 00:01:18.413
So when we scale vector U by
three, we could just do this

00:01:18.413 --> 00:01:23.119
as three times vector U, we
know is a vector two, comma,

00:01:23.119 --> 00:01:28.006
negative one, and so that's
going to be, we could write it

00:01:28.006 --> 00:01:30.936
this way, so let me write
the two, let me write

00:01:30.936 --> 00:01:35.914
the negative one, that's
going to be three times two,

00:01:37.554 --> 00:01:39.853
for the new X component
once we scale it up.

00:01:39.853 --> 00:01:44.797
And three times negative
one for the new Y component,

00:01:44.797 --> 00:01:48.388
and of course that's going
to be, result in the vector

00:01:48.388 --> 00:01:51.766
three times two, our new X
component is six, and our new

00:01:51.766 --> 00:01:55.969
Y component is going to be negative three.

00:01:55.969 --> 00:01:57.155
And so let's plot that.

00:01:57.155 --> 00:01:59.080
So everything I've done just now

00:01:59.080 --> 00:02:02.586
is that part of the expression,
that part of the expression.

00:02:02.586 --> 00:02:04.548
So the vector six, comma, negative three,

00:02:04.548 --> 00:02:08.971
if we start it at the origin,
we're going to move, let's see

00:02:08.971 --> 00:02:13.948
we're going to move one,
two, three, four, five, six

00:02:15.160 --> 00:02:19.717
in the X direction and negative
three in the Y direction.

00:02:19.717 --> 00:02:23.754
So one, two, three,

00:02:23.754 --> 00:02:26.903
one, two, three,

00:02:26.903 --> 00:02:31.398
so we would get right about there.

00:02:31.398 --> 00:02:35.414
So there you have it, this
is the vector three U.

00:02:35.414 --> 00:02:38.479
And we're at one, two,
three, four, five, six

00:02:38.479 --> 00:02:41.335
in the X direction, or six to the right,

00:02:41.335 --> 00:02:45.038
and then we went down
three, one, two, three,

00:02:45.038 --> 00:02:48.393
in the Y direction, and
notice, it's in the exact same

00:02:48.393 --> 00:02:52.572
direction as vector U, it just
has three times the magnitude

00:02:52.572 --> 00:02:56.136
that's vector U, that would be two U.

00:02:56.136 --> 00:02:59.793
And then we get to three U,
three times the magnitude.

00:02:59.793 --> 00:03:03.694
Alright, now let's figure out
what one fifth times W is.

00:03:03.694 --> 00:03:07.072
Let's see one fifth times W,
well let me just write this,

00:03:07.072 --> 00:03:11.992
it's gonna be plus one fifth,
W is the vector, is the vector

00:03:12.772 --> 00:03:17.626
negative five, comma, five,
and so this is gonna be plus,

00:03:19.299 --> 00:03:23.362
so that's gonna be, let me
write the components down.

00:03:23.362 --> 00:03:28.282
So it's gonna be plus one fifth, times,

00:03:28.282 --> 00:03:31.478
one fifth times negative five,

00:03:31.478 --> 00:03:35.194
and the Y component is gonna
be one fifth times five,

00:03:35.194 --> 00:03:38.305
and so that's gonna be plus,
whoops I wrote one half,

00:03:38.305 --> 00:03:40.592
my brains not working properly,

00:03:40.592 --> 00:03:45.572
one fifth times negative five,
and one fifth times five.

00:03:45.572 --> 00:03:50.007
And so, this part right over
here is going to be one fifth

00:03:50.007 --> 00:03:53.792
times negative five, is negative one,

00:03:53.792 --> 00:03:56.880
and one fifth times five is positive one.

00:03:56.880 --> 00:03:59.573
And so this new vector, one fifth W,

00:03:59.573 --> 00:04:02.080
is gonna be W scaled down.

00:04:02.080 --> 00:04:04.147
And so it's negative one,
comma, one if we start it

00:04:04.147 --> 00:04:07.258
at the origin, so
negative one, comma, one.

00:04:07.258 --> 00:04:09.429
We'd get right over there, and notice,

00:04:09.429 --> 00:04:12.004
it's going in the same direction as W,

00:04:12.004 --> 00:04:15.255
it's just one fifth, it's
just one fifth as long.

00:04:15.255 --> 00:04:17.902
And now we just wanna
add these two vectors.

00:04:17.902 --> 00:04:20.119
So if we add them by just
looking at its components,

00:04:20.119 --> 00:04:23.695
the resulting vector,
the resulting vector,

00:04:23.695 --> 00:04:27.537
let me do this in a new color
that I have not used yet,

00:04:27.537 --> 00:04:31.182
so the resulting vector, we're
gonna add the corresponding

00:04:31.182 --> 00:04:33.501
X components, so it's going to be

00:04:33.501 --> 00:04:37.738
six plus negative one,

00:04:37.738 --> 00:04:41.836
six plus negative one, and
the resulting Y component

00:04:41.836 --> 00:04:46.619
is going to be negative three plus one.

00:04:46.619 --> 00:04:51.158
Negative three plus one.

00:04:51.158 --> 00:04:55.844
And so the resulting vector
is going to be equal to

00:04:55.844 --> 00:05:00.793
five, comma, negative two.

00:05:00.793 --> 00:05:03.141
And we can also see that visually.

00:05:03.141 --> 00:05:06.322
If we start with this blue
vector, three times the vector U,

00:05:06.322 --> 00:05:10.257
and we were to add the
green vector, one fifth W,

00:05:10.257 --> 00:05:14.204
well we were to add that we
would just start at the head of

00:05:14.204 --> 00:05:18.256
three U, then we're gonna
add negative one, comma, one,

00:05:18.256 --> 00:05:20.856
so we're gonna move one
to the left and one up,

00:05:20.856 --> 00:05:22.934
we're gonna get right over there.

00:05:22.934 --> 00:05:25.527
So let me see if I can draw that.

00:05:25.527 --> 00:05:27.675
So, just gonna do a
little head to tail method

00:05:27.675 --> 00:05:30.194
right over here, so the
head of the first vector

00:05:30.194 --> 00:05:31.390
is gonna be where the tail of the

00:05:31.390 --> 00:05:33.134
next vector starts, that we're adding.

00:05:33.134 --> 00:05:36.698
And so the resulting vector
is going to be, if we started

00:05:36.698 --> 00:05:41.539
at the tail of the first
vector, three U right over here,

00:05:41.539 --> 00:05:45.288
or at the origin, and then
we bring it to the head of

00:05:45.288 --> 00:05:48.748
the second vector, we get, once again,

00:05:48.748 --> 00:05:52.451
the vector five, comma, negative two.

00:05:52.451 --> 00:05:54.866
Its X component is five, we're
gonna move five to the right,

00:05:54.866 --> 00:05:58.075
and we move two down.


Adding and subtracting vectors

WEBVTT
Kind: captions
Language: en

00:00:00.329 --> 00:00:01.765
- [Voiceover] So let's get
some practice and hopefully

00:00:01.765 --> 00:00:04.006
a little bit of intuition
for adding and subtracting

00:00:04.006 --> 00:00:06.544
two dimensional vectors.

00:00:06.544 --> 00:00:10.225
So let's say that I have vector A, and

00:00:10.225 --> 00:00:14.646
let's say that its x component is

00:00:14.646 --> 00:00:19.126
I don't know, three, and it's
y component is negative one.

00:00:19.126 --> 00:00:23.105
And let's say we also have a vector B.

00:00:23.105 --> 00:00:26.523
Vector B, and let's say
that its x component

00:00:26.523 --> 00:00:31.388
is two, and let's say
its y component is three.

00:00:31.388 --> 00:00:35.298
Now, let's first think about what would

00:00:35.298 --> 00:00:38.496
A plus B equal?

00:00:38.496 --> 00:00:41.176
So what resulting vector would this be?

00:00:41.176 --> 00:00:44.939
A plus B is equal to what?

00:00:44.939 --> 00:00:46.678
And I encourage you to pause the video

00:00:46.678 --> 00:00:49.174
and think about what this would be.

00:00:49.174 --> 00:00:51.147
Well, the convention
is, is if we're taking

00:00:51.147 --> 00:00:53.060
the sum of two vectors, we can just add up

00:00:53.060 --> 00:00:55.900
their x, their x components to get our new

00:00:55.900 --> 00:00:57.733
x component and up their y components

00:00:57.733 --> 00:00:59.340
to get the new y component.

00:00:59.340 --> 00:01:02.739
So the x component of
vector A plus vector B

00:01:02.739 --> 00:01:06.799
is going to be three plus two.

00:01:06.799 --> 00:01:09.515
That's going to be the x
component, which we know is five.

00:01:09.515 --> 00:01:12.894
And the y component is
going to be negative one.

00:01:12.894 --> 00:01:16.755
Negative one plus three.

00:01:16.755 --> 00:01:21.435
Negative one plus three,
and so the resulting vector

00:01:21.435 --> 00:01:23.479
is going to have an x component

00:01:23.479 --> 00:01:28.479
of five and a y component
of negative one plus three

00:01:28.537 --> 00:01:31.339
is equal to two.

00:01:31.339 --> 00:01:33.079
Alright, that was pretty straightforward.

00:01:33.079 --> 00:01:34.657
Now instead of adding
them, let's think about

00:01:34.657 --> 00:01:36.700
what would happen if we subtract them.

00:01:36.700 --> 00:01:39.798
So let's think about what A, what vector A

00:01:39.798 --> 00:01:43.618
minus vector B would be,
and I suspect that you might

00:01:43.618 --> 00:01:46.475
be able to guess or at least think about

00:01:46.475 --> 00:01:49.260
what would happen if you subtract vector B

00:01:49.260 --> 00:01:52.475
instead of adding vector B.

00:01:52.475 --> 00:01:56.116
Well, as you might have
guessed, instead of

00:01:56.116 --> 00:01:59.415
adding the corresponding
components, we subtract it.

00:01:59.415 --> 00:02:01.202
So our x component is going to be the

00:02:01.202 --> 00:02:05.097
x component of A minus
the x component of B

00:02:05.097 --> 00:02:07.958
'cause it's vector A minus vector B.

00:02:07.958 --> 00:02:10.694
So our x component is going to be

00:02:10.694 --> 00:02:12.936
three minus two.

00:02:12.936 --> 00:02:17.116
Three minus two,

00:02:17.116 --> 00:02:19.975
and our y component is going to be

00:02:19.975 --> 00:02:23.685
the y component of A, negative one,

00:02:23.685 --> 00:02:26.088
minus the y component of B.

00:02:26.088 --> 00:02:28.691
So minus three.

00:02:28.691 --> 00:02:32.591
Minus three, and so the resulting vector

00:02:32.591 --> 00:02:36.752
is going to be, is going to
be the vector three minus two.

00:02:36.752 --> 00:02:38.173
The x component's going to be one,

00:02:38.173 --> 00:02:40.247
and then the y component of negative one

00:02:40.247 --> 00:02:43.581
minus three is negative four.

00:02:43.581 --> 00:02:45.239
Now what I just showed
you, this is the convention

00:02:45.239 --> 00:02:47.939
for adding and subtracting
two dimensional vectors

00:02:47.939 --> 00:02:49.439
like vectors A and B.

00:02:49.439 --> 00:02:51.002
Let's think a little bit about how we can

00:02:51.002 --> 00:02:54.136
visually depict what is going on.

00:02:54.136 --> 00:02:58.138
So let's first visually depict
adding vector A and vector B.

00:02:58.138 --> 00:03:02.422
So let me draw some axes, and so my,

00:03:02.422 --> 00:03:07.422
so this could be, there's my y-axis.

00:03:07.838 --> 00:03:09.700
Y-axis, let's see.

00:03:09.700 --> 00:03:11.956
The highest y value I get to is,

00:03:11.956 --> 00:03:14.256
the highest y value I get to is three,

00:03:14.256 --> 00:03:15.858
and the lowest is negative four.

00:03:15.858 --> 00:03:19.282
And then, so let me draw
the x-axis some place.

00:03:19.282 --> 00:03:21.196
Whoops, didn't mean to do that.

00:03:21.196 --> 00:03:22.882
I did the zoom by accident.

00:03:22.882 --> 00:03:24.118
Alright.

00:03:24.118 --> 00:03:27.436
So the x-axis, put it right over there.

00:03:29.096 --> 00:03:31.307
There's my x-axis, and let's see.

00:03:31.307 --> 00:03:34.107
The highest x value is five.

00:03:34.107 --> 00:03:35.271
The lowest is three.

00:03:35.271 --> 00:03:36.929
Actually, I could just focus on the,

00:03:36.929 --> 00:03:40.886
I could just focus on
the, let me just focus

00:03:40.886 --> 00:03:42.585
on the right hand side.

00:03:42.585 --> 00:03:45.541
So that's y, and then let me do

00:03:45.541 --> 00:03:50.240
x looking like that.

00:03:50.240 --> 00:03:51.964
X-axis.

00:03:51.964 --> 00:03:52.845
And now let's see.

00:03:52.845 --> 00:03:56.720
Vector A is 3, -1.

00:03:56.720 --> 00:04:01.720
So one, two, three, and negative one.

00:04:02.384 --> 00:04:04.800
If we want to just visualize
it in standard form,

00:04:04.800 --> 00:04:06.945
we could put its initial
point at the origin

00:04:06.945 --> 00:04:09.865
and its terminal point at the point 3, -1.

00:04:09.865 --> 00:04:12.500
And so we could draw it like that.

00:04:12.500 --> 00:04:14.043
We could also shift it
around, as long as it has

00:04:14.043 --> 00:04:16.524
the same, as long as it
has the same magnitude

00:04:16.524 --> 00:04:20.141
and direction, we could shift it around.

00:04:20.141 --> 00:04:23.265
So that right there is our vector A.

00:04:23.265 --> 00:04:26.879
Now Vector B, 2,3, we could
have its initial point

00:04:26.879 --> 00:04:30.683
at the origin and just, we
could draw it like this.

00:04:30.683 --> 00:04:32.859
So the x-coordinate is two.

00:04:32.859 --> 00:04:36.242
Y-coordinate one, two, three.

00:04:36.242 --> 00:04:38.425
So its terminal point could be there.

00:04:38.425 --> 00:04:42.799
So we could just draw it
like this for vector B,

00:04:42.799 --> 00:04:45.539
but if we're going to
add vector B to vector A

00:04:45.539 --> 00:04:48.001
like we have right over
there, what we want to do is

00:04:48.001 --> 00:04:51.919
shift vector B over so that
its, I guess you could say,

00:04:51.919 --> 00:04:55.101
tail starts at A's head,
or its initial point

00:04:55.101 --> 00:04:57.462
starts at A's terminal point.

00:04:57.462 --> 00:04:58.920
So let's do that.

00:04:58.920 --> 00:05:01.838
So if we start right over
here, we're going to go

00:05:01.838 --> 00:05:03.519
two in the x direction.

00:05:03.519 --> 00:05:06.223
So we're going to go two
more in the x direction,

00:05:06.223 --> 00:05:08.104
and we're going to go
three in the y direction.

00:05:08.104 --> 00:05:10.841
So one, two, three.

00:05:10.841 --> 00:05:13.025
So we're going to end up right over there.

00:05:13.025 --> 00:05:15.645
So notice, this is the
same vector, vector B.

00:05:15.645 --> 00:05:17.725
I've just shifted it over.

00:05:17.725 --> 00:05:19.778
Has the same magnitude and same direction

00:05:19.778 --> 00:05:21.442
as what I had drawn before,

00:05:21.442 --> 00:05:23.197
as that vector right over there.

00:05:23.197 --> 00:05:25.622
I have just shifted it
down into the right.

00:05:25.622 --> 00:05:27.997
And now we do this, so this is vector A,

00:05:27.997 --> 00:05:29.123
I'm adding Vector B.

00:05:29.123 --> 00:05:31.542
I put the, I put the
initial point of vector B

00:05:31.542 --> 00:05:33.400
at the terminal point of vector A.

00:05:33.400 --> 00:05:35.163
I've shifted it over right over there.

00:05:35.163 --> 00:05:37.420
So now I can figure out
the resulting vector,

00:05:37.420 --> 00:05:41.190
A plus B, the resulting vector A plus B,

00:05:41.190 --> 00:05:43.161
by going from the initial point of A

00:05:43.161 --> 00:05:45.719
to the terminal point of B.

00:05:45.719 --> 00:05:48.798
So it's gonna start here,
and then go over there,

00:05:48.798 --> 00:05:52.282
and notice, this vector I'm
going five in the x direction,

00:05:52.282 --> 00:05:54.067
and two in the y direction.

00:05:54.067 --> 00:05:58.480
This is the vector 5,2 right over there.

00:05:58.480 --> 00:06:01.017
That is vector A plus vector B.

00:06:01.017 --> 00:06:03.742
But also think about what
happens when we do A minus B.

00:06:03.742 --> 00:06:06.323
Well, we could still have
vector A just like we drew it,

00:06:06.323 --> 00:06:10.441
but now instead of
putting vector B's tail or

00:06:10.441 --> 00:06:13.637
initial point at the
terminal point of vector A,

00:06:13.637 --> 00:06:15.160
we would wanna put negative B.

00:06:15.160 --> 00:06:18.067
So negative B would just
be, have the same magnitude

00:06:18.067 --> 00:06:20.441
but it'll be in the opposite direction.

00:06:20.441 --> 00:06:23.337
So instead of going two
to the right and three up,

00:06:23.337 --> 00:06:27.922
we would go two to the
left and three down.

00:06:27.922 --> 00:06:32.922
So just two the left and
one, two, three down,

00:06:33.384 --> 00:06:36.739
and we would end up right about there.

00:06:36.739 --> 00:06:39.438
And so, if you subtract A minus B,

00:06:39.438 --> 00:06:42.180
that's the same thing
as A plus negative B.

00:06:42.180 --> 00:06:45.800
So negative B is going to
look something like this.

00:06:45.800 --> 00:06:48.883
It's going to be the
exact opposite direction.

00:06:48.883 --> 00:06:52.478
It's gonna look like that, and so there

00:06:52.478 --> 00:06:55.638
if you start at the
initial point of A and get

00:06:55.638 --> 00:06:59.538
to the terminal point of the
negative B now, this is all

00:06:59.538 --> 00:07:01.637
hand-drawn, so it could be
a little bit more precise,

00:07:01.637 --> 00:07:06.637
notice we get to the point 1, -4.

00:07:07.017 --> 00:07:07.920
1, -4.

00:07:07.920 --> 00:07:09.658
So this vector right
over here, let me do that

00:07:09.658 --> 00:07:12.999
in a different color, this brown, whoops,

00:07:12.999 --> 00:07:14.038
that's not what I wanted to do.

00:07:14.038 --> 00:07:17.341
This brown vector right
over here, that is the

00:07:17.341 --> 00:07:20.877
vector A minus B, and then this white one,

00:07:20.877 --> 00:07:23.038
actually, let me do this
in another color as well,

00:07:23.038 --> 00:07:27.582
this magenta vector right
here, that is A plus B.

00:07:27.582 --> 00:07:28.562
So hopefully that makes sense,

00:07:28.562 --> 00:07:30.282
and if you, given the
components, you're just, if

00:07:30.282 --> 00:07:32.462
you're adding the vectors, add
the correspondent components.

00:07:32.462 --> 00:07:34.500
If you're subtracting the
vectors, well if you're

00:07:34.500 --> 00:07:37.341
subtracting B from A,
subtract B's components,

00:07:37.341 --> 00:07:38.343
corresponding components from

00:07:38.343 --> 00:07:40.590
the corresponding components of A.


Visually adding and subtracting vectors

WEBVTT
Kind: captions
Language: en

00:00:00.271 --> 00:00:03.112
- [Voiceover] Let's build our
intuition for visually adding

00:00:03.112 --> 00:00:05.152
and subtracting vectors.

00:00:05.152 --> 00:00:09.552
So let's say that I have
vector a, and I add that

00:00:09.552 --> 00:00:13.292
to vector, to vector b,

00:00:13.292 --> 00:00:17.154
and the resulting vector is a vector c,

00:00:17.154 --> 00:00:19.514
is vector c.

00:00:19.514 --> 00:00:21.634
So what could this look like visually

00:00:21.634 --> 00:00:25.034
if we assume that a, b and c
are two-dimensional vectors?

00:00:25.034 --> 00:00:28.414
Well, I'm just going to draw
what vector a might look like.

00:00:28.414 --> 00:00:31.762
So let's say that this
right over here is vector a,

00:00:32.696 --> 00:00:36.346
and vector b, vector b,

00:00:36.976 --> 00:00:38.658
since I'm adding it to
vector a, I'm going to put

00:00:38.658 --> 00:00:43.098
its initial point at the
terminal point of vector a,

00:00:43.098 --> 00:00:44.978
and then I'm going to draw vector b.

00:00:44.978 --> 00:00:48.618
So let's say vector b looks like that.

00:00:48.618 --> 00:00:51.260
So that is vector b, let me label these.

00:00:51.260 --> 00:00:55.660
That is vector a, this is vector b.

00:00:55.660 --> 00:00:58.580
And I did that so that I can
figure out what the sum is,

00:00:58.580 --> 00:01:00.700
what vector c is going to be.

00:01:00.700 --> 00:01:03.240
So that is vector b,
and what would that be?

00:01:03.240 --> 00:01:08.098
Well, we would start at the
initial point of vector a

00:01:08.950 --> 00:01:12.250
and then go to the
terminal point of vector b.

00:01:12.250 --> 00:01:15.152
So this right over there

00:01:15.152 --> 00:01:17.392
would be the sum.

00:01:17.392 --> 00:01:21.094
So that would be vector
c right over there.

00:01:21.094 --> 00:01:24.814
So the important realization
is if I add two vectors,

00:01:24.814 --> 00:01:28.934
I would put the tail of one
at the head of the other.

00:01:28.934 --> 00:01:30.976
And now what's neat
about this if I'm adding,

00:01:30.976 --> 00:01:32.454
the order doesn't matter.

00:01:32.454 --> 00:01:33.976
I could've done this the other way around.

00:01:33.976 --> 00:01:35.496
I could've started with vector b.

00:01:35.496 --> 00:01:39.622
I could've said vector b plus vector a

00:01:39.622 --> 00:01:43.784
is equal to vector c,
is equal to vector c,

00:01:43.784 --> 00:01:45.024
and you could see that visually.

00:01:45.024 --> 00:01:47.984
It would be a slightly
different visual diagram

00:01:47.984 --> 00:01:49.824
but you get to the same place.

00:01:49.824 --> 00:01:53.304
So if I start with vector b,
let's say I start over here,

00:01:54.274 --> 00:01:55.628
In fact, you don't have
to start at the origin

00:01:55.628 --> 00:01:57.268
but let's say that was the origin.

00:01:57.268 --> 00:02:00.988
So I could start with
vector b, draw vector b

00:02:00.988 --> 00:02:05.128
just like that, and
then add vector a to it.

00:02:05.128 --> 00:02:09.168
So start vector a at the
terminal point of vector b,

00:02:09.168 --> 00:02:12.648
and then go to (mumbles)
just draw vector a.

00:02:12.648 --> 00:02:13.772
So vector a.

00:02:14.150 --> 00:02:16.330
So once again, a vector,
I can shift them around

00:02:16.330 --> 00:02:17.594
as long as I'm not changing the direction

00:02:17.594 --> 00:02:19.332
or their magnitude.

00:02:19.332 --> 00:02:22.054
So vector a looks like that.

00:02:22.054 --> 00:02:24.724
And notice, if you now go and
start at the initial point

00:02:24.724 --> 00:02:27.952
of b and go to the terminal
point of a, you still get

00:02:27.952 --> 00:02:29.072
vector c.

00:02:29.072 --> 00:02:32.792
So that's why a plus b and b
plus a are going to give you

00:02:32.792 --> 00:02:34.752
the same thing.

00:02:34.752 --> 00:02:37.112
Now what if instead of saying a plus b

00:02:37.112 --> 00:02:40.992
I wanted to think about what
a minus b is going to be?

00:02:40.992 --> 00:02:42.482
So let me write that down.

00:02:42.482 --> 00:02:47.472
Vector a minus vector b, minus vector b.

00:02:48.094 --> 00:02:50.814
And let's call that vector d.

00:02:50.814 --> 00:02:53.834
That is equal to vector d.

00:02:53.834 --> 00:02:56.194
So once again, I could start with vector a

00:02:56.194 --> 00:02:57.794
and here, order matters.

00:02:57.794 --> 00:03:01.794
So vector a looks something like this.

00:03:01.794 --> 00:03:03.714
this is hand-drawn so it's not going to be

00:03:03.714 --> 00:03:05.434
completely perfect.

00:03:05.434 --> 00:03:07.314
So vector a, just like that.

00:03:07.314 --> 00:03:09.954
And one way of thinking
about subtracting vector b

00:03:09.954 --> 00:03:12.474
is instead of adding vector
b the way we did here,

00:03:12.474 --> 00:03:15.114
we could add negative b.

00:03:15.114 --> 00:03:17.114
So negative b would
have the same magnitude

00:03:17.114 --> 00:03:19.434
but just the opposite direction.

00:03:19.434 --> 00:03:20.914
So that's vector a.

00:03:20.914 --> 00:03:24.194
Vector negative b will
still start right over here,

00:03:24.194 --> 00:03:26.394
but will go in the opposite direction.

00:03:26.394 --> 00:03:27.794
So let's do that.

00:03:27.794 --> 00:03:31.114
So negative b is going to look like this,

00:03:31.114 --> 00:03:33.726
is going to look something like this.

00:03:35.016 --> 00:03:37.016
So that is negative b.

00:03:37.016 --> 00:03:41.576
Notice, same magnitude
exactly opposite direction.

00:03:41.576 --> 00:03:43.476
We've flipped it around 180 degrees,

00:03:43.476 --> 00:03:46.598
and now the resulting
vector is going to be d.

00:03:46.598 --> 00:03:50.758
So vector d is going to look like that,

00:03:51.960 --> 00:03:54.240
vector d.

00:03:54.240 --> 00:03:58.530
So c is a plus b, d is a minus b.

00:03:58.530 --> 00:04:03.510
Or you can even call this
a plus, a plus negative b.

00:04:04.162 --> 00:04:07.662
Now with that out of the
way, let's draw some diagrams

00:04:07.662 --> 00:04:08.584
and go the other way.

00:04:08.584 --> 00:04:10.104
See if we could go from the diagrams

00:04:10.104 --> 00:04:12.724
to the actual equations.

00:04:12.724 --> 00:04:16.384
So let's start with, let
me draw an interesting one.

00:04:16.384 --> 00:04:18.474
So let's say that this...

00:04:19.744 --> 00:04:22.268
Let's say that's vector a.

00:04:22.268 --> 00:04:23.918
Vector a.

00:04:26.388 --> 00:04:27.728
I'll use green.

00:04:27.728 --> 00:04:32.488
Let's say that, that is vector b,

00:04:32.488 --> 00:04:35.458
and I will now use magenta.

00:04:35.850 --> 00:04:39.980
And let's say that this is vector c.

00:04:40.810 --> 00:04:41.970
Vector c.

00:04:41.970 --> 00:04:43.670
So I encourage you to pause the video

00:04:43.670 --> 00:04:46.130
and see if you can write an equation

00:04:46.130 --> 00:04:48.950
that defines this relationship.

00:04:48.950 --> 00:04:51.170
Well, this is interesting
because they're all going

00:04:51.170 --> 00:04:53.070
in a circle right over here.

00:04:57.322 --> 00:04:59.764
Let's say that you started at...

00:04:59.764 --> 00:05:01.124
This is your initial point.

00:05:01.124 --> 00:05:03.844
You said, okay, a plus b,

00:05:03.844 --> 00:05:07.004
well, if you're trying to
figure out what a plus b

00:05:07.004 --> 00:05:10.744
is going to be, the resulting
vector would start here

00:05:10.744 --> 00:05:11.864
and end here.

00:05:11.864 --> 00:05:14.464
But vector c is going in
the opposite direction.

00:05:14.464 --> 00:05:17.284
But we could, instead of thinking
about vector c like this,

00:05:17.284 --> 00:05:20.024
we could think about the
opposite of vector c,

00:05:20.024 --> 00:05:21.584
which would do,

00:05:21.584 --> 00:05:24.064
so instead of calling this
c, I could flip this around

00:05:24.064 --> 00:05:25.984
by calling it this negative c.

00:05:25.984 --> 00:05:29.416
So I could flip this around,

00:05:30.286 --> 00:05:33.526
and now, let me do the same color,

00:05:33.526 --> 00:05:37.256
this would be equal to negative c.

00:05:38.046 --> 00:05:40.246
Notice, before I just had vector c here

00:05:40.246 --> 00:05:43.646
and it started at this point
and ended at this point.

00:05:43.646 --> 00:05:45.886
Now I just flipped it
around, it has the exact

00:05:45.886 --> 00:05:47.606
opposite direction, same magnitude.

00:05:47.606 --> 00:05:48.966
Now it is negative c.

00:05:48.966 --> 00:05:51.092
And this makes it easier for
us to construct an equation

00:05:51.092 --> 00:05:54.012
because negative c starts at
the tip, at the initial point

00:05:54.012 --> 00:05:58.232
or the tail of vector a,
and it goes to the head

00:05:58.232 --> 00:06:00.832
of vector b or the
terminal point of vector b.

00:06:00.832 --> 00:06:02.612
So we can now write an equation.

00:06:02.612 --> 00:06:06.612
We could say vector a plus vector b,

00:06:06.612 --> 00:06:09.694
plus vector b,

00:06:09.694 --> 00:06:13.336
is equal to, is equal to not c,

00:06:13.336 --> 00:06:17.456
it's equal to the negative of vector c.

00:06:17.456 --> 00:06:19.986
So hopefully, you found that interesting.


Examples from understanding scalar multiplication exercise

WEBVTT
Kind: captions
Language: en

00:00:00.451 --> 00:00:02.684
- [Voiceover] All right,
so we're told that vector v

00:00:02.684 --> 00:00:04.985
is equal to x comma y.

00:00:04.985 --> 00:00:07.752
They're not giving us the actual numbers

00:00:07.752 --> 00:00:09.684
for the x component and the y component.

00:00:09.684 --> 00:00:11.084
They're just saying that the x component

00:00:11.084 --> 00:00:12.751
is going to be the variable x.

00:00:12.751 --> 00:00:15.651
And the y component is
going to be the variable y.

00:00:15.651 --> 00:00:20.219
And the magnitude of vector v
is going to be equal to five.

00:00:20.219 --> 00:00:22.951
All right, I think I
can digest all of that.

00:00:22.951 --> 00:00:24.587
And it says, fill in
the blanks to complete

00:00:24.587 --> 00:00:26.717
the following sentences.

00:00:26.717 --> 00:00:31.184
So they're saying vector
w right over here,

00:00:31.184 --> 00:00:33.418
they didn't have to actually
write the word vector.

00:00:33.418 --> 00:00:35.117
They have an arrow on top of the w,

00:00:35.117 --> 00:00:36.351
so we know it's a vector.

00:00:36.351 --> 00:00:41.351
So vector w its components
are three x and three y.

00:00:42.118 --> 00:00:44.918
So its x component is
three times the x component

00:00:44.918 --> 00:00:46.017
of vector v.

00:00:46.017 --> 00:00:49.285
And its y component is
three times the y component

00:00:49.285 --> 00:00:50.885
of vector v.

00:00:50.885 --> 00:00:52.452
Instead of having a y, it has a three y.

00:00:52.452 --> 00:00:54.785
Instead of having an x, it has a three x.

00:00:54.785 --> 00:00:58.551
So w is essentially
vector v being multiplied

00:00:58.551 --> 00:01:00.785
by the scalar three.

00:01:00.785 --> 00:01:03.521
So its magnitude is
going to be three times

00:01:03.521 --> 00:01:06.918
the magnitude of vector v.

00:01:06.918 --> 00:01:09.953
So if vector v has a magnitude of five,

00:01:09.953 --> 00:01:14.818
vector w's going to
have a magnitude of 15.

00:01:14.818 --> 00:01:17.585
It's going to have three
times the magnitude.

00:01:17.585 --> 00:01:20.418
All right, now they tell
us vector z is equal to,

00:01:20.418 --> 00:01:23.452
so its x component is negative two times

00:01:23.452 --> 00:01:26.185
the x component of vector v.

00:01:26.185 --> 00:01:29.785
And its y component is negative
two times the y component

00:01:29.785 --> 00:01:31.553
of vector v.

00:01:31.553 --> 00:01:36.553
So you could view vector z as
negative two times vector v.

00:01:37.218 --> 00:01:41.751
Or the scalar negative
two times the vector v.

00:01:41.751 --> 00:01:44.351
And so you might say okay
that means its magnitude

00:01:44.351 --> 00:01:46.886
is going to be negative two
times the magnitude of v.

00:01:46.886 --> 00:01:48.351
But you have to remember, magnitude,

00:01:48.351 --> 00:01:50.051
that's always going to be positive.

00:01:50.051 --> 00:01:51.484
One way to think, when we talk about even

00:01:51.484 --> 00:01:52.751
absolute value we're really thinking about

00:01:52.751 --> 00:01:55.252
the magnitude of something.

00:01:55.252 --> 00:01:56.851
We're talking about how long it is.

00:01:56.851 --> 00:01:58.618
We're not thinking about its direction.

00:01:58.618 --> 00:02:01.687
So we wouldn't think
about the negative there.

00:02:01.687 --> 00:02:02.718
If we think about its direction,

00:02:02.718 --> 00:02:04.984
this negative is going to flip
its direction and we could

00:02:04.984 --> 00:02:06.718
think about that in a second.

00:02:06.718 --> 00:02:08.185
But if we just think about magnitude,

00:02:08.185 --> 00:02:09.952
we care about the two.

00:02:09.952 --> 00:02:11.885
So its going to have
two times the magnitude

00:02:11.885 --> 00:02:13.551
of vector v.

00:02:13.551 --> 00:02:15.621
So instead of five, it's
going to be five times two,

00:02:15.621 --> 00:02:16.851
which is ten.

00:02:16.851 --> 00:02:18.751
So now they say, match
each vector to defined

00:02:18.751 --> 00:02:21.284
to a vector that could be its equivalent.

00:02:21.284 --> 00:02:22.753
So if we think about it, the vector with

00:02:22.753 --> 00:02:25.452
the smallest magnitude
is going to be vector v.

00:02:25.452 --> 00:02:29.984
The other two we're
multiplying vector v by scalars

00:02:29.984 --> 00:02:32.484
with an absolute value of more than one.

00:02:32.484 --> 00:02:34.151
So they're going to
have larger magnitudes.

00:02:34.151 --> 00:02:36.850
So the smallest magnitude
is this one right over here.

00:02:36.850 --> 00:02:38.250
This is the shortest vector.

00:02:38.250 --> 00:02:40.819
So that would be our vector v.

00:02:40.819 --> 00:02:44.018
Now, vector w would be
one that has three times

00:02:44.018 --> 00:02:47.317
the magnitude and it's
going in the same direction.

00:02:47.317 --> 00:02:50.318
So, this one has three
times the magnitude.

00:02:50.318 --> 00:02:52.284
It looks like three times, roughly.

00:02:52.284 --> 00:02:54.618
And it's definitely going
in the same direction.

00:02:54.618 --> 00:02:57.185
And then vector z would have
two times the magnitude.

00:02:57.185 --> 00:02:58.651
But since we have, we're multiplying it

00:02:58.651 --> 00:03:00.518
by the negative scalar,

00:03:00.518 --> 00:03:02.118
it's going to go in
the opposite direction.

00:03:02.118 --> 00:03:03.118
And this looks like it.

00:03:03.118 --> 00:03:05.417
This purple arrow looks like
it's about twice the length

00:03:05.417 --> 00:03:06.584
of this red arrow.

00:03:06.584 --> 00:03:08.618
And it's going in the opposite direction.

00:03:08.618 --> 00:03:12.051
So I feel pretty good about
what we have done here.

00:03:12.051 --> 00:03:14.051
Let's do one more of these.

00:03:14.051 --> 00:03:17.785
So, let's do this one here.

00:03:17.785 --> 00:03:20.285
Here we have a vector v again.

00:03:20.285 --> 00:03:23.418
It has a magnitude of 10.

00:03:23.418 --> 00:03:24.751
And it says, fill in
the blanks to complete

00:03:24.751 --> 00:03:25.891
the following sentences.

00:03:25.891 --> 00:03:30.891
Vector w now, so now
vector w, it's taking each

00:03:30.951 --> 00:03:33.718
of the components of vector
v and multiplying them

00:03:33.718 --> 00:03:35.784
by one-fifth.

00:03:35.784 --> 00:03:40.350
So you can view vector w as
being equal to one-fifth,

00:03:40.350 --> 00:03:44.251
the scalar one-fifth times vector v.

00:03:44.251 --> 00:03:47.485
The magnitude of w is
going to be one-fifth

00:03:47.485 --> 00:03:49.551
the magnitude of v.

00:03:49.551 --> 00:03:52.850
One-fifth of 10 is just going to be two.

00:03:52.850 --> 00:03:56.317
Now vector z, it looks
like it's three-fifths

00:03:56.317 --> 00:03:57.818
times vector v.

00:03:57.818 --> 00:04:00.317
We're multiplying each of the
components times three-fifths.

00:04:00.317 --> 00:04:03.085
So that's like multiplying
the entire vector

00:04:03.085 --> 00:04:04.784
by the scalar three-fifths.

00:04:04.784 --> 00:04:07.318
So it's magnitude is
going to be three-fifths

00:04:07.318 --> 00:04:09.217
the magnitude of vector v.

00:04:09.217 --> 00:04:12.751
And so three-fifths of 10 is six.

00:04:12.751 --> 00:04:16.552
All right, then they say,
match each vector defined above

00:04:16.552 --> 00:04:18.351
to a vector that could be its equivalent.

00:04:18.351 --> 00:04:21.117
So here the vector with
the largest magnitude

00:04:21.117 --> 00:04:22.618
is vector v.

00:04:22.618 --> 00:04:25.051
This would be vector v.

00:04:25.051 --> 00:04:26.886
And the vector with the smallest magnitude

00:04:26.886 --> 00:04:28.418
is vector w.

00:04:28.418 --> 00:04:30.819
It's one-fifth the magnitude.

00:04:30.819 --> 00:04:33.385
So this has a magnitude, should be 10.

00:04:33.385 --> 00:04:35.752
This should one one-fifth of that

00:04:35.752 --> 00:04:37.720
which is a magnitude of
two and it looks like that.

00:04:37.720 --> 00:04:40.818
And this should be three-fifths of the 10,

00:04:40.818 --> 00:04:42.084
which is a magnitude of six.

00:04:42.084 --> 00:04:43.161
And that looks about right,

00:04:43.161 --> 00:04:45.284
so I feel good with
what we have filled out.

00:04:45.284 --> 00:04:48.018
So hopefully that gives you
a sense of how to tackle

00:04:48.018 --> 00:04:51.902
these types of example problems.


Understanding multiplying vectors by scalars

WEBVTT
Kind: captions
Language: en

00:00:00.177 --> 00:00:01.273
- [Voiceover] What I
wanna do in this video is

00:00:01.273 --> 00:00:04.667
give ourselves some practice,
and hopefully some intuition,

00:00:04.667 --> 00:00:08.202
on multiplying a scalar times a vector.

00:00:08.202 --> 00:00:09.403
Now, what am I talking about when I say,

00:00:09.403 --> 00:00:12.104
multiplying a scalar times a vector?

00:00:12.104 --> 00:00:14.539
Well, let me set up a little
two-dimensional vector here.

00:00:14.539 --> 00:00:18.228
Let's say I have the vector w,

00:00:18.228 --> 00:00:20.388
and let me give it an x component.

00:00:20.388 --> 00:00:23.996
Let's say its x component is one

00:00:23.996 --> 00:00:28.295
and its y component
is, let's say it's two.

00:00:28.295 --> 00:00:29.594
And I could draw it if I like.

00:00:29.594 --> 00:00:30.569
Actually, that's a good idea.

00:00:30.569 --> 00:00:32.662
It's always nice to be able
to visualize these things.

00:00:32.662 --> 00:00:36.419
Let me get some coordinate axis here.

00:00:36.941 --> 00:00:40.379
So, that's my x-axis,

00:00:40.379 --> 00:00:42.295
that is my y-axis.

00:00:42.295 --> 00:00:45.241
And so, if I were to draw this vector

00:00:45.241 --> 00:00:47.539
in standard form, I would
put its initial point

00:00:47.539 --> 00:00:48.634
at the origin.

00:00:48.634 --> 00:00:50.094
And then its terminal point would be

00:00:50.094 --> 00:00:52.384
at the point one comma two.

00:00:52.384 --> 00:00:54.761
So its x coordinate is one,

00:00:54.761 --> 00:00:57.829
its y coordinate is going to be two.

00:00:57.829 --> 00:00:59.861
So, one, two.

00:00:59.861 --> 00:01:03.118
This vector is going to look like,

00:01:03.714 --> 00:01:06.131
its initial point is right here,

00:01:06.131 --> 00:01:08.894
its terminal point is going
to be right over there,

00:01:08.894 --> 00:01:11.989
the vector, in standard,

00:01:12.632 --> 00:01:14.278
graphing it in standard form

00:01:14.278 --> 00:01:16.007
or visualizing it in standard
form, would look like that.

00:01:16.007 --> 00:01:18.010
Now, of course, I could
have the same vector

00:01:18.010 --> 00:01:19.837
and I could shift it
around as long as I have

00:01:19.837 --> 00:01:21.402
the same length of the arrow

00:01:21.402 --> 00:01:23.231
and it's pointing in the same direction.

00:01:23.231 --> 00:01:26.094
But if no one tells you
otherwise, it's nice

00:01:26.094 --> 00:01:29.426
to just put its initial
point at the origin.

00:01:29.426 --> 00:01:31.844
Now, let's multiply it by a scalar.

00:01:31.844 --> 00:01:33.339
What do we mean by a scalar?

00:01:33.339 --> 00:01:34.692
Well, a vector is something that has

00:01:34.692 --> 00:01:36.773
a magnitude and a direction.

00:01:36.773 --> 00:01:38.775
A scalar is just something
that has a magnitude.

00:01:38.775 --> 00:01:41.274
You could think of just the numbers

00:01:41.274 --> 00:01:43.145
that you started learning
when you were four years old,

00:01:43.145 --> 00:01:45.074
those are scalars.

00:01:45.074 --> 00:01:46.943
So, for example, we could think about,

00:01:46.943 --> 00:01:50.205
what is three

00:01:50.205 --> 00:01:52.177
times w

00:01:52.177 --> 00:01:53.598
going to be?

00:01:53.598 --> 00:01:55.711
Three times w.

00:01:55.711 --> 00:01:57.337
Three is a scalar,

00:01:57.337 --> 00:01:59.572
w is a vector.

00:01:59.572 --> 00:02:02.076
Now, the convention we use
for multiplying a scalar

00:02:02.076 --> 00:02:04.983
times a vector is, you just
multiply each of the components

00:02:04.983 --> 00:02:07.035
times that scalar.

00:02:07.035 --> 00:02:09.207
So this is going to be equal to,

00:02:11.255 --> 00:02:13.043
we have a one and a two,
and we're gonna multiply

00:02:13.043 --> 00:02:15.684
each of those times the three.

00:02:15.684 --> 00:02:17.046
Three times one,

00:02:17.046 --> 00:02:18.773
and then three times two,

00:02:18.773 --> 00:02:21.683
and so this is going to be equal to,

00:02:21.683 --> 00:02:23.551
this is going to be equal to,

00:02:23.551 --> 00:02:25.165
three times one is three,

00:02:25.165 --> 00:02:27.765
three times two is six.

00:02:27.765 --> 00:02:30.407
And so we see the resulting vector,

00:02:31.207 --> 00:02:33.318
we could call this vector three w,

00:02:33.318 --> 00:02:35.233
it's gonna have an x component of three

00:02:35.233 --> 00:02:36.879
and a y component of six.

00:02:36.879 --> 00:02:40.143
So if I were to draw it
in standard form here,

00:02:40.143 --> 00:02:43.739
x component one, two, three,

00:02:44.200 --> 00:02:46.821
and then y component

00:02:46.821 --> 00:02:49.523
two, three, four,

00:02:49.523 --> 00:02:52.420
five and six.

00:02:52.420 --> 00:02:55.487
And so, it's going to look like this.

00:02:55.487 --> 00:02:58.303
Let me see if I can draw it reasonably.

00:02:58.303 --> 00:03:02.014
It's going to look like this.

00:03:02.014 --> 00:03:03.455
And obviously, I'm hand-drawing it,

00:03:03.455 --> 00:03:05.913
so it's not going to be exactly right.

00:03:05.913 --> 00:03:07.762
But it's going to look like that.

00:03:07.762 --> 00:03:12.333
So this is the vector three times w.

00:03:12.333 --> 00:03:13.572
Now, notice what happened

00:03:13.572 --> 00:03:15.665
when I multiplied it by the scalar.

00:03:15.665 --> 00:03:18.936
The direction didn't change,
but the magnitude did.

00:03:18.936 --> 00:03:20.907
And you see what the magnitude changed by.

00:03:20.907 --> 00:03:24.381
It increased by a factor of three.

00:03:25.059 --> 00:03:27.307
The length of my blue arrow now

00:03:27.307 --> 00:03:31.437
is three times the length
of my magenta arrow.

00:03:31.437 --> 00:03:32.536
Let's do another example.

00:03:32.536 --> 00:03:35.257
I'll use the same vector w,
since we already have it set up.

00:03:35.257 --> 00:03:37.817
Let's multiply it times a negative scalar.

00:03:37.817 --> 00:03:39.865
Let's say, let's see what
happens if I multiply

00:03:39.865 --> 00:03:41.531
negative two

00:03:41.531 --> 00:03:44.246
times w.

00:03:44.246 --> 00:03:45.691
And I (mumbles) a positive.

00:03:45.691 --> 00:03:46.871
You don't think about what this would be.

00:03:46.871 --> 00:03:49.301
And even, if you have
the time, graph it out.

00:03:49.882 --> 00:03:51.753
Well, we would multiply
each of our components

00:03:51.753 --> 00:03:53.499
by negative two.

00:03:53.499 --> 00:03:56.831
So it would be equal to negative two

00:03:56.831 --> 00:03:58.070
times one,

00:03:58.070 --> 00:03:59.187
would be the x component,

00:03:59.187 --> 00:04:00.366
and then the y component would be

00:04:00.366 --> 00:04:02.966
negative two times two.

00:04:02.966 --> 00:04:06.136
And so, this is going to
be equal to the vector

00:04:06.136 --> 00:04:09.906
negative two comma negative four.

00:04:09.906 --> 00:04:11.348
Well, let's plot that.

00:04:11.348 --> 00:04:12.914
The x component is negative two.

00:04:12.914 --> 00:04:15.414
So it's negative one, negative two.

00:04:15.414 --> 00:04:16.604
And the y component,

00:04:16.604 --> 00:04:19.529
negative one, two,

00:04:19.529 --> 00:04:22.312
three, I'm going a little
bit off of my axes,

00:04:22.312 --> 00:04:24.528
four, so that would be
negative four there,

00:04:24.528 --> 00:04:25.929
that's negative two.

00:04:25.929 --> 00:04:28.584
So it's going to look something like this.

00:04:28.584 --> 00:04:32.958
It's going to look something like,

00:04:32.958 --> 00:04:35.470
something like that.

00:04:35.871 --> 00:04:37.166
So that right over there is the vector

00:04:37.166 --> 00:04:39.501
negative two w.

00:04:39.501 --> 00:04:41.370
Now let's think about what just happened.

00:04:41.370 --> 00:04:44.377
Well, because we had the negative here,

00:04:44.377 --> 00:04:47.458
it essentially flipped in
the direction by 180 degrees.

00:04:47.458 --> 00:04:49.701
It's going in the opposite direction.

00:04:49.701 --> 00:04:51.005
But one way to think about it,

00:04:51.005 --> 00:04:53.748
they kind of would still
sit on the same line.

00:04:53.748 --> 00:04:55.881
So the negative just
flipped its direction.

00:04:55.881 --> 00:04:59.355
If you consider whatever
direction this was,

00:04:59.355 --> 00:05:01.121
the magenta vector, w, was going,

00:05:01.121 --> 00:05:03.275
it's now going to go in
the opposite direction.

00:05:03.275 --> 00:05:06.202
And then it also scaled it up by two.

00:05:06.202 --> 00:05:08.415
This is twice as long,
has twice the magnitude

00:05:08.415 --> 00:05:09.614
of our original vector,

00:05:09.614 --> 00:05:11.301
and it's going in the opposite direction

00:05:11.301 --> 00:05:13.088
because of the negative sign.

00:05:13.088 --> 00:05:14.694
So hopefully this gives you
a little bit on intuition

00:05:14.694 --> 00:05:17.334
of what it means to scale a vector.

00:05:17.334 --> 00:05:20.145
And literally, the word scalar,

00:05:21.005 --> 00:05:22.401
let me write it down.

00:05:23.201 --> 00:05:24.961
Scalar.

00:05:24.961 --> 00:05:26.465
That's a scalar, that's a scalar.

00:05:26.465 --> 00:05:27.968
It has the word scale in it.

00:05:27.968 --> 00:05:30.447
It's literally just scaling the vector.

00:05:30.447 --> 00:05:32.621
It is changing its magnitude.

00:05:32.621 --> 00:05:34.957
It might flip it around
because of a negative sign,

00:05:34.957 --> 00:05:37.070
but it's essentially
changing its magnitude,

00:05:37.070 --> 00:05:39.205
scaling it up or down
or flipping it around

00:05:39.205 --> 00:05:41.078
with a negative sign.


Finding vector magnitude from components

WEBVTT
Kind: captions
Language: en

00:00:00.203 --> 00:00:01.501
- [Voiceover] Let's do some examples

00:00:01.501 --> 00:00:03.182
figuring out the magnitude of a vector

00:00:03.182 --> 00:00:05.472
if we're just given some
information about it.

00:00:05.472 --> 00:00:07.399
So, one of the simplest cases would be

00:00:07.399 --> 00:00:09.982
well, if they just told
us the actual components

00:00:09.982 --> 00:00:10.870
of the vector.

00:00:10.870 --> 00:00:14.604
So if they said vector a is equal to,

00:00:14.604 --> 00:00:18.695
let's say five comma negative three,

00:00:18.695 --> 00:00:21.443
this means that its
x-component is positive five,

00:00:21.443 --> 00:00:23.970
its y-component is negative three.

00:00:23.970 --> 00:00:27.190
Well, if we have this,
then the magnitude of a,

00:00:27.190 --> 00:00:30.281
the magnitude of a is just going to be,

00:00:30.281 --> 00:00:32.966
and this really just comes
from the distance formula

00:00:32.966 --> 00:00:35.315
which just comes from
the Pythagorean theorem,

00:00:35.315 --> 00:00:36.991
the magnitude of a is just going to be

00:00:36.991 --> 00:00:38.265
the square root

00:00:38.265 --> 00:00:42.355
of the x-component squared.

00:00:42.355 --> 00:00:44.552
So let me do that in a different color.

00:00:44.552 --> 00:00:47.322
So the square root of
the x-component squared,

00:00:47.322 --> 00:00:49.655
so five squared,

00:00:49.655 --> 00:00:53.315
plus the y-component squared,

00:00:53.315 --> 00:00:56.538
so plus negative three squared.

00:00:56.538 --> 00:00:59.209
And this is going to be equal to

00:00:59.209 --> 00:01:04.188
the square root of 25,

00:01:04.898 --> 00:01:08.303
25 plus nine,

00:01:08.303 --> 00:01:10.313
plus nine,

00:01:10.313 --> 00:01:13.501
which is equal to the square root of 34,

00:01:13.501 --> 00:01:18.392
which is equal to the square root of 34.

00:01:18.492 --> 00:01:19.998
And if you want to think
about this visually,

00:01:19.998 --> 00:01:21.029
this is very easy to do

00:01:21.029 --> 00:01:22.796
just looking at the actual components.

00:01:22.796 --> 00:01:24.072
But if you want to make sense of this,

00:01:24.072 --> 00:01:27.190
why this is essentially just
the Pythagorean theorem,

00:01:27.190 --> 00:01:30.739
we could draw out a quick
coordinate axis right over here.

00:01:30.739 --> 00:01:33.499
So that's our y-axis.

00:01:33.499 --> 00:01:35.221
This is our, let's see,

00:01:35.221 --> 00:01:37.051
I have a y-component of negative three.

00:01:37.051 --> 00:01:38.037
So let's see.

00:01:38.037 --> 00:01:41.257
That is our, actually let me
draw it a little bit different.

00:01:41.257 --> 00:01:42.522
Let me draw it like this.

00:01:42.522 --> 00:01:44.718
That is our x-axis.

00:01:44.718 --> 00:01:48.198
And we see its x-component
is positive five,

00:01:48.198 --> 00:01:52.926
so one, two, three, four, five.

00:01:52.926 --> 00:01:54.546
That's five there.

00:01:54.546 --> 00:01:56.884
And its y-component is negative three.

00:01:56.884 --> 00:02:00.254
So one, two, three.

00:02:00.254 --> 00:02:02.454
And so this is negative three.

00:02:02.454 --> 00:02:05.915
And so we can draw this
vector with its initial point.

00:02:05.915 --> 00:02:07.588
Remember, we can always
shift around a vector

00:02:07.588 --> 00:02:09.849
as long as we don't change its
magnitude and to direction.

00:02:09.849 --> 00:02:11.665
We can start it at the origin,

00:02:11.665 --> 00:02:14.484
and make it go five in the x-direction

00:02:14.484 --> 00:02:17.208
and negative three in the y-direction,

00:02:17.208 --> 00:02:19.439
and so its terminal point
will be right over there

00:02:19.439 --> 00:02:22.027
at the point five comma negative three.

00:02:22.027 --> 00:02:24.883
And so the vector, the vector,

00:02:24.883 --> 00:02:27.874
will look like this.

00:02:27.874 --> 00:02:29.353
And if we want to figure
out the magnitude,

00:02:29.353 --> 00:02:31.719
that's just the length of this line.

00:02:31.719 --> 00:02:34.817
And what we can do is just
set up a right triangle

00:02:34.817 --> 00:02:38.960
where our change, our change in y

00:02:38.960 --> 00:02:43.015
is this negative three right over here.

00:02:43.015 --> 00:02:44.422
That is our change in y.

00:02:44.422 --> 00:02:48.881
And our change in x is this positive five,

00:02:48.881 --> 00:02:50.366
is that positive five.

00:02:50.366 --> 00:02:51.941
And so this is a right triangle.

00:02:51.941 --> 00:02:54.035
Five squared plus,

00:02:54.035 --> 00:02:55.403
you could just view the absolute value

00:02:55.403 --> 00:02:56.473
of this side as three,

00:02:56.473 --> 00:02:58.544
so five squared plus three squared

00:02:58.544 --> 00:03:00.826
is going to be the hypotenuse squared.

00:03:00.826 --> 00:03:04.153
Comes straight out of
the Pythagorean theorem.