[ Music ]
>> Hello everyone. Welcome back to Engineering
Mechanics. Our discussion will focus on vectors
today. The concept of vector is very important
for all engineers. You're going to repeatedly
use the concept of vectors in solving various
engineering problems. Some of you might have
heard about vectors already in high school,
and you might have heard definitions such
as vector is a physical quantity that has
magnitude and direction. Now, if that definition
puzzled you, I want you to know the concept
of vector is very simple and easy. In fact,
believe it or not, you have been using the
concept of vector in your everyday life. Let
me give you an example. If you ever give a
direction to someone to get to a place, and
if you give directions along the lines that
I am going to say, for example, "Go forward,
go in this direction 100 feet, take a left,
then go again forward for 20 or 30 feet. And
the place that you're looking for is on your
left." If you give such directions, you are
using the concept of vector. You are telling
someone how far to go, in which direction
to go. That is actually dealing with vectors.
Give you another example. Say you are driving
from Los Angeles to Las Vegas. So you are
driving west at 50 miles an hour in your car.
If you say that, that you are driving towards
west at 50 miles an hour, you are dealing
with the concept of vector because you are
giving a number related to the speed such
as 50 miles, and then you are also giving
a direction. That's vector. Or if you play
sports, you are kicking a soccer ball towards
the goal, you are kicking at certain angle
and you are applying certain amount of force.
That is dealing with vectors. Or you are shooting
a basketball into a hoop. That means you are
throwing a basketball at an angle and you
are applying just the right amount of force
so it can go into the hoop. You are dealing
with vectors. Alright? So vector is really
a concept that includes two important parameters.
One is a value, which I am going to use the
word to describe as magnitude, which is actually
a number. So a value, a magnitude, a number,
a vector has that, as well as direction. Which
way? East, west, maybe 30 degrees from the
horizontal, may be at an angle of 45 degrees.
These two concepts come together, gives you
vector. So vector is a more generic name.
You can -- a force is a vector because you
are applying certain amount of force such
as 50 pounds, 100 pounds along a direction;
that's a vector. Or velocity is a vector.
Going in certain direction at certain speed.
Acceleration is a vector. So there are many,
many engineering concepts, engineering quantities
that can be called a vector. So vector is
an important concept that basically allows
us to deal with many engineering quantities
easily, and it allows us to help perform mathematical
operations. So in the next few minutes, we
are going to take a closer look at vectors,
learn how to add them, subtract them, and
deal with them in many different ways. So
let's take a few examples. Alright, everybody.
Let's take a closer look at vectors. We talked
about vectors a while ago. And we know vectors
are quantities, physical quantities that have
magnitude and direction. As you know magnitude
is simply a number. It could be something
like 10, 76, and so on. It is just a number.
It's a value. A direction is, for example,
going this way. In this case, it is going
from, say, east to west. In fact, I gave you
an example of going from, say, Los Angeles
to Las Vegas, right? That will be going east
at certain speed which is like 50 miles an
hour. That will be a vector. So any physical
quantity, engineering quantity with magnitude
and direction can be called a vector. What
are the examples of vector? Well, we just
talked about it, which is a position vector,
right? Position with respect to something
is a vector. A vector is typically denoted
by an arrow at certain angle. So that's a
position. Similarly velocity, acceleration,
these are all examples of vectors. Well, there
is one more example I want to spend a few
more minutes. It is the force. Alright. Let's
take a look at force vector. Force is an important
concept and we are going to be dealing with
this concept all the time in statics. So let's
take a force vector. Force, as all of you
know, is a push or pull. It's like pushing
something or pulling something. If it is push
or pull, it has certain direction, right?
I am pulling it this way or I am pushing it
this way. So force has direction. Plus we
also apply certain amount of force. If you
are using American system, you may say I am
applying 50 pound force. Or you may be using
metric system, you may say I am going to spend
or I'm going to exert 100 Newton force. So
force has certain amount which is the value
magnitude and the direction. So force is a
vector. Now, we have an understanding of what
a vector is. Vector is a quantity with direction
and magnitude. Now the next question is, how
do we represent the vector? How do we show
a vector? Well, take a look at it this way.
We want to show a vector using an arrow. This
indicates which way the sense is this way.
And also it makes an angle, in this particular
case, from the horizontal, say, angle theta.
So f I want to denote a vector, typically
you will have to show some kind of an angle
as well as the direction. And to write this
down, let's say this is a force vector. I'm
going to put a little dash on top of it. In
books, you may find a bold letter for vector
indicating it is not just an f, it is vector
f. So vector f can be written like this. And
it may be equal to, for example, 50 pounds
magnitude making certain angle along the horizontal
-- from the horizontal. In this case, theta.
Now, typically in engineering, we measure
the angle from the x axis, from the horizontal.
So this is a conventional way of providing
the vector's magnitude and direction, alright?
And there are other ways of representing;
for now let's stick to this. So anytime you
are dealing with a vector, your answer must
include two parts. One is the magnitude -- in
this particular case, 50 pounds -- as well
as the direction it makes with a horizontal
or an x axis in this case. Also the sense
should be shown properly with an arrow, alright?
If you miss an arrow, it is not a vector.
In engineering, we want to speak a common
language across the world. It doesn't matter
what language you speak; if you represent
a vector, people from different countries
speaking different languages will understand.
So it's very important you follow the notations.
So this is the vector notation, and please
follow this and do not ignore any one of those
parts, alright? So now that we know how to
represent a vector, we know what a vector
is, the next topic is on how can we manipulate
these vectors? And why do we have to do that?
Think about it. You are applying some force,
say you want to pull something. It is too
heavy for you to pull in so you ask someone
else to help. So let's say you have this big
furniture. Two of us are pulling. We tie a
rope. I pull this way, the other person pulls
that way. So each one of us is applying certain
amount of force and pulling it in a direction
with the hope this big piece of furniture
is going to move towards us, alright? So two
people are pulling, and what is going to be
the net effect of these two people pulling
that furniture? So what we have to now do
is to figure out the way to add the force,
add two forces actually. And this will be
adding two vectors, because forces are vectors
and we want to add two forces, alright? So
let's talk about how can we add two forces,
or in other words, how can we add two vectors.
Let's take a look at vector addition. So vector
addition. As you know, a vector can be denoted
simply something like this. Let's say I'm
going to call this vector p, alright? Notice
how I am writing it. Now let's say here is
another vector, q, alright? And I want to
add these two vectors. There are couple of
ways of doing this. One is the graphical approach.
A graphical approach is basically based on
making a sketch to the scale. So you need
proper drafting tools such as triangles, rulers,
protractors and a pencil. So assuming you
have all that, let's do a graphical approach
of adding two vectors. So given two vectors,
what we want to do is to figure out a way
to draw graphically the resultant of these
two vectors. Remember, two people are pulling
something. I am pulling it this way; someone
else is pulling it this way. These are two
force vectors, and this is going to have a
net effect or a resultant vector somewhere
in this direction along this way. But I do
not know exactly what that is, and in order
to do that, I'm going to draw a simple graphical
approach called parallelogram approach. In
other words, I'm going to draw two parallel
lines to these vectors. So what I do is, I'm
going to draw a line parallel to vector p
from the head -- this is the head of a vector,
this is the tail of the vector. I'm going
to draw a line parallel. Similarly, I'm going
to draw a line parallel to vector q from the
head close, okay? So this is parallel to this;
this is parallel to that. Now, these two vectors,
the lines intersect, and this point is where
the resultant's head is and the resultant
is going to look something like this. This
is my resultant vector. So vector addition
can be done graphically by using what I call
as a parallelogram approach. And remember,
in order to solve vector problems like this,
you must draw them to scale. Once you have
drawn to scale, you can measure the length
of this vector and multiply here your scaling
value, and that will be the magnitude. In
order to get the direction, assuming this
is your horizontal line, you can measure this
angle and the resultant vector is r. So this
is how you add vectors using parallelogram
method by drawing parallel lines. Now, there
is also another approach, which is effectively
the parallelogram approach, but sometimes
people use a different word -- triangle. And
what does it mean? Well, basically you draw
a triangle instead of a parallelogram, but
you get the same answer. Here, a graphical
approach, here is a vector horizontal, here
is a vector, p and q, and in parallelogram
law we drew two parallel lines, and the intersection
gave us the resultant. Now, using triangle
law, all we are doing is, here is my original
vector p, alright? Now you can transfer this
vector q here, which is effectively drawing
a parallel line again to that, and these two
endpoints will give you the resultant. So
effectively, parallelogram law or triangle
law both are effectively the same but people
use different terms. I want to make sure you
understand they're effectively the same. You
can use any word that you are comfortable
with. So this is a graphical approach. Once
you have found the resultant, you're going
to express the answer using the magnitude
and the direction. Please do not forget that.
So you can measure these angles, alright?
Now, graphical approach is very simple, easy
to use. I encourage you to explore this. Having
said that, it is also not very effective because
you have to draw them to scale, which means
it is not going to be very accurate. So sometimes
or often, we would like to find out other
ways of solving it. One way to solve that
will be using an analytical approach. An analytical
approach will involve formulas. Instead of
just drawing to scale, you just make a sketch,
use the information to calculate the information
more accurately. So let's take a look at a
different way of adding vectors, alright?
Before we do that, I also want to make sure
that you know how to subtract vectors graphically
because, you know, adding and subtracting
are effectively the same operations. Instead
of adding two vectors, you take another vector,
add the negative of that, and that will be
subtraction. If you're not clear, let me show
you a very quick example. Vector subtraction.
So again, let me draw the same example. In
this case, we had vector p here, this is vector
q, p plus q, vector p plus vector q is your
r. This is how we added vectors and we found
the resultant using parallelogram or triangle
law, and this is your r, alright? Now let's
say we want to find out p minus r. How do
we subtract? Well, p is your first vector;
it's right here. Remember my q vector originally
is like this. Now I need to subtract. So all
I have to do is to reverse the direction.
So instead of going this way, I am going to
come this way. This is my p now, this is my
negative q, alright, because it is in the
negative direction. So p minus vector q would
give you a resultant. You see that? So this
is the result of adding; this is the result
of subtracting. So what we have done is, we
are adding the corresponding negative vector.
Instead of adding q, we are subtracting q,
in this case, alright? So subtraction is pretty
straightforward and easy, and I hope you understand
that. And now let's take a quick look at multiplying
a vector by a scalar. Now, I am using the
word scalar; I am not sure if I used it before.
A scalar means it is just a number. We talked
about the word magnitude, right? Scalar is
just a magnitude; it does not have any direction.
What are the examples of scalar? Well, area
of a triangle. It doesn't have a direction.
It is just a number. How many people live
in California? It's a number. It doesn't have
a direction. So anytime you use a number -- your
age, the population, area of a geometric figure
-- these are all magnitudes or scalars. So
let's say we take a scalar such as 5 and multiply
vector p. So this is my vector p originally,
meaning it's a certain magnitude and direction.
Now, if I multiply vector p by a scalar, it
becomes 5p, and therefore it will become five
times longer, alright? So this is 5p. In other
words, what has changed from here to here
is not the direction -- only the magnitude.
So if you take a vector and multiply by a
scalar, the vector changes magnitude by those
many times, but the direction remains the
same. So multiplying a vector by a scalar
is a very simple process. If your force vector
was, say, 50 pounds, let's say here is a force
vector, and this force vector is equal to
50 pounds at this direction along x axis now
if I say 5f, it will become five times larger
like that, alright? Which means it is going
to be 250 pounds in its magnitude, alright?
I think it's pretty straightforward and simple.
By the way, dividing a vector by a scalar
will be multiplying by its reciprocal, so
it's pretty much the same operation as multiplication
so I'm not going to spend time on it, alright?
I hope you understand this. These are all
simple approaches, adding and subtraction.
I have shown you a graphical approach. Now
I would like to move on to an analytical approach
with an example.
