We can never have too much practice getting familiar with vectors.
What I want to do in this video in particular,
be able to relate the x and y components of a vector
to the initial and terminal points to where we place it
in the coordinate planes.
What am I talking about?
Let's say we have some vector where its initial point is here at point a
and its terminal point is here at point b.
The vector will look like this in green.
So that's what the vector looks like.
It has an initial point at a and the terminal point at b.
What I want to do is, if I know the coordinates of a and b
figure out the x and y components of my vector v.
So what are the coordinates of a?
The x coordinate is -1,-2,-3,-4,-5,-6,-7,-8.
So a = (-8,
and the y coordinate is 1,2,3,4,5,6,7.
So a=(-8, 7)
and point b, the terminal point of the vector, has the coordinates...
1,2,3,4,5,6,7. b=(7,
and then the y coordinate is -1,-2.
b=(7, -2).
So, given that, what are going to be the x and y components
of our vector v ?
Well, let's think about it, our vector v.
There are multiple ways that we can express a vector,
one way is to use the unit vector notation,
where whatever is in parentheses is our x component,
and whatever is in these parentheses is our y component.
I'm really just scaling up the unit vector in the horizontal direction
and the unit vector in the vertical direction.
You could look at the videos on unit vector notation
if this looks foreign to you.
But let's figure out what the x component is going to be.
One way to think about it is, you could just view it as the change in x
between the initial point and the terminal point.
So the change in x is going to be
the terminal x, 7 minus the initial x, -8
and our y component you can just view as a change in y
from the initial point to the terminal point.
So we finish at the terminal point,
that is -2 minus our initial y, -7.
And so what does this give us?
7 minus -8, that's the same as 7 plus 8,
that's equal to 15.
So 15 times i plus -2 minus 7, that's -9,
or -9j. That's one way to write our vector.
Another way is to just remember there are a bunch of different notations
people use for vectors.
You could write it like this; you could just say it's 15, -9.
These are different representations of the same vector.
You could even write it like this, vector v, you could write it as
a column vector, 15, -9.
So depending on what book or notation or class you're in,
you might see all of these things,
but these are all the same vectors that you know and love.
We can see that over here,
15 i is going to be a vector that's 15 times bigger than the unit vector
in the horizontal direction.
So 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15.
So this is the vector 15 i.
And the vector -9 j is going to go down 9,
so 1,2,3,4,5,6,7,8,9.
So -9 j is going to look like this.
And you see if you take 15 i plus -9 j,
you're going to get our entire vector v,
which is exactly what we wrote here.
You could say the x component is 15, the y component is -9.
Or another way to say it is, to go from point a to point b,
your change in x, you increased 15,
your change in y, you decreased 9.
That tells you the x and y components
of the vector that starts here and ends there.
This just happens to be where we placed this vector,
we might choose to shift them around, but the important thing is
that they keep the same magnitude and direction.