- [Voiceover] So let's get some practice and hopefully
a little bit of intuition for adding and subtracting
two dimensional vectors.
So let's say that I have vector A, and
let's say that its x component is
I don't know, three, and it's y component is negative one.
And let's say we also have a vector B.
Vector B, and let's say that its x component
is two, and let's say its y component is three.
Now, let's first think about what would
A plus B equal?
So what resulting vector would this be?
A plus B is equal to what?
And I encourage you to pause the video
and think about what this would be.
Well, the convention is, is if we're taking
the sum of two vectors, we can just add up
their x, their x components to get our new
x component and up their y components
to get the new y component.
So the x component of vector A plus vector B
is going to be three plus two.
That's going to be the x component, which we know is five.
And the y component is going to be negative one.
Negative one plus three.
Negative one plus three, and so the resulting vector
is going to have an x component
of five and a y component of negative one plus three
is equal to two.
Alright, that was pretty straightforward.
Now instead of adding them, let's think about
what would happen if we subtract them.
So let's think about what A, what vector A
minus vector B would be, and I suspect that you might
be able to guess or at least think about
what would happen if you subtract vector B
instead of adding vector B.
Well, as you might have guessed, instead of
adding the corresponding components, we subtract it.
So our x component is going to be the
x component of A minus the x component of B
'cause it's vector A minus vector B.
So our x component is going to be
three minus two.
Three minus two,
and our y component is going to be
the y component of A, negative one,
minus the y component of B.
So minus three.
Minus three, and so the resulting vector
is going to be, is going to be the vector three minus two.
The x component's going to be one,
and then the y component of negative one
minus three is negative four.
Now what I just showed you, this is the convention
for adding and subtracting two dimensional vectors
like vectors A and B.
Let's think a little bit about how we can
visually depict what is going on.
So let's first visually depict adding vector A and vector B.
So let me draw some axes, and so my,
so this could be, there's my y-axis.
Y-axis, let's see.
The highest y value I get to is,
the highest y value I get to is three,
and the lowest is negative four.
And then, so let me draw the x-axis some place.
Whoops, didn't mean to do that.
I did the zoom by accident.
Alright.
So the x-axis, put it right over there.
There's my x-axis, and let's see.
The highest x value is five.
The lowest is three.
Actually, I could just focus on the,
I could just focus on the, let me just focus
on the right hand side.
So that's y, and then let me do
x looking like that.
X-axis.
And now let's see.
Vector A is 3, -1.
So one, two, three, and negative one.
If we want to just visualize it in standard form,
we could put its initial point at the origin
and its terminal point at the point 3, -1.
And so we could draw it like that.
We could also shift it around, as long as it has
the same, as long as it has the same magnitude
and direction, we could shift it around.
So that right there is our vector A.
Now Vector B, 2,3, we could have its initial point
at the origin and just, we could draw it like this.
So the x-coordinate is two.
Y-coordinate one, two, three.
So its terminal point could be there.
So we could just draw it like this for vector B,
but if we're going to add vector B to vector A
like we have right over there, what we want to do is
shift vector B over so that its, I guess you could say,
tail starts at A's head, or its initial point
starts at A's terminal point.
So let's do that.
So if we start right over here, we're going to go
two in the x direction.
So we're going to go two more in the x direction,
and we're going to go three in the y direction.
So one, two, three.
So we're going to end up right over there.
So notice, this is the same vector, vector B.
I've just shifted it over.
Has the same magnitude and same direction
as what I had drawn before,
as that vector right over there.
I have just shifted it down into the right.
And now we do this, so this is vector A,
I'm adding Vector B.
I put the, I put the initial point of vector B
at the terminal point of vector A.
I've shifted it over right over there.
So now I can figure out the resulting vector,
A plus B, the resulting vector A plus B,
by going from the initial point of A
to the terminal point of B.
So it's gonna start here, and then go over there,
and notice, this vector I'm going five in the x direction,
and two in the y direction.
This is the vector 5,2 right over there.
That is vector A plus vector B.
But also think about what happens when we do A minus B.
Well, we could still have vector A just like we drew it,
but now instead of putting vector B's tail or
initial point at the terminal point of vector A,
we would wanna put negative B.
So negative B would just be, have the same magnitude
but it'll be in the opposite direction.
So instead of going two to the right and three up,
we would go two to the left and three down.
So just two the left and one, two, three down,
and we would end up right about there.
And so, if you subtract A minus B,
that's the same thing as A plus negative B.
So negative B is going to look something like this.
It's going to be the exact opposite direction.
It's gonna look like that, and so there
if you start at the initial point of A and get
to the terminal point of the negative B now, this is all
hand-drawn, so it could be a little bit more precise,
notice we get to the point 1, -4.
1, -4.
So this vector right over here, let me do that
in a different color, this brown, whoops,
that's not what I wanted to do.
This brown vector right over here, that is the
vector A minus B, and then this white one,
actually, let me do this in another color as well,
this magenta vector right here, that is A plus B.
So hopefully that makes sense,
and if you, given the components, you're just, if
you're adding the vectors, add the correspondent components.
If you're subtracting the vectors, well if you're
subtracting B from A, subtract B's components,
corresponding components from
the corresponding components of A.