Let's say we have the vector a and we know its x and y components.
I'll write it first as a column vector.
It's x component is -10 and its y component is +7.
And with the many ways we've seen to represent a vector
we could also represent it in unit vector form,
so we could write it as -10
times the unit vector in the horizontal direction
plus 7 times the unit vector in the vertical direction.
But either way we could draw this thing,
we could pick a place, an initial point.
If we say the initial point is the origin,
we might as well pick the origin,
what would this vector look like?
Well, the x component is -10, so we'd go to -10,
and then the y component is 7.
So we do 10 to the left and 7 up.
7 up would get us to right over there, and the vector would look like this.
So it would start at the origin and would end,
its initial point would be the origin and its terminal point
would be at x= -10 and y= 7.
So this is one way to represent the vector a.
But let's say we wanted to move it around,
we wanted to have a different terminal point.
Let's say we were told
instead of having the terminal point -10, 7
let's say instead it had the terminal point 3,5.
Let's make this a negative 3.
-3, 5, let's say it had that as its terminal point.
So we could imagine what the vector would look like.
We would just translate this vector over so it would look something like this.
So it would terminate right over there,
and it would start somewhere in this area here.
I don't know precisely, if I could just copy/paste this
and translate it over, I could figure it out on the graph.
But instead of doing it like that let's do it mathematically.
What would be the initial point of this vector, of vector a
if the terminal point is -3, 5?
Let's say the initial point I is equal to x Initial, y Initial.
So x sub i, y sub i.
So let's say that's it.
How do we go from having an initial point and a terminal point
to getting the components of the vector?
The x component of the vector is going to be the change in x
going from the initial to the terminal point
and your y component is going to be your change in y
going from your initial point to your terminal point.
So -10, your x component is going to be your change in x
going from your initial to your terminal.
That's going to be your terminal x, -3,
minus your initial x sub i.
Then we can do the same thing for the y component.
The y component is going to be equal to your change in y.
It's going to be your terminal y, it's going to be 5,
minus your initial y.
And from this we can solve for x sub i, your initial x,
and your initial y.
So let's see how we can do this.
If we multiplied both sides of this times -1,
we're going to get 10 is equal to x i +3
and if we subtract 3 from both sides,
we're going to get x sub i is equal to 7.
x sub i is equal to 7.
Now let's do the same thing over here.
If we multiplied both sides of this times -1,
we're going to have -7 is equal to y sub i.
Our Initial y minus 5.
Add 5 to both sides, our Initial y is equal to -2.
So our initial point for our vector, if our terminal point is going to be -3,5
based on the components of the vector, our initial point is going to be
at point 7, -2.
Let's see if that makes visual sense.
This is x= 7,-2.
So we're saying that is our initial point,
so our vector is going to look like this,
which makes a lot of sense.
It's just the original vector we had
when we had the original point of the origin,
but we've shifted it over,
and shifted it to the right, 1,2,3,4,5,6,7.
And you see that right over here.
Hopefully now we're pretty well-equipped.
If we know the initial and terminal point of a vector
we can figure out its x and y components.
Or if we know the terminal points and the x and y components,
we can figure out the initial point.
Or if we know the initial point and we know the x and y components
we can figure out the terminal point.
So if we know two out of these three,
we can figure out the third piece of information.