>> Welcome to a lesson on vector projection.
The goal of this video is to determine the projection of one vector on to another.
So if we want to project vector U on to vector V we use this notation here.
Vector U is being projected on to vector V. Looking at the graph here
on the left we're projecting the red vector U on to the blue vector V
and the projection is the gray vector which we call vector W sub 1
so to emphasize this the projection of the red vector
on to the blue vector is this gray vector here.
One way to think of a projection of one vector on to another is if we were to have a light
up here and it was shining down on the red vector the projection of U
on to V would be the shadow that this light produces.
We can also think of the projection as the vector component of U along V. Looking
at the second diagram if we had a light up here and projected vector U
on to vector V this light would produce a shadow in the opposite direction of vector V. Again,
the projection would be this gray vector here, vector W sub 1.
So the projection doesn't necessarily have to be in the same direction
but it would be a scalar multiple of vector V. The next thing I want to mention is
that vector W sub 2 is called the vector component of U orthogonal to V
and we can find W-2 by taking vector U and subtracting vector W sub 1.
So again the highlighted gray vector would be the vector component of U along V
and the vector component of U orthogonal
to V would be vector W sub 2 would look something like this.
Again, notice that the green vector is orthogonal to the blue vector
and vector W 1 plus vector W-2 does equal vector U. On the second sketch here vector W sub 2
or the vector component of U that's orthogonal to vector V would be this vector here.
Again, notice it's orthogonal or perpendicular to vector V
and vector W 1 plus vector W 2 would equal vector U. Now let's go ahead and take a look
at a formula that will allow us to find the projection of U on to V. If U
and V are nonzero vectors so that projection of U on to V is given by this formula here notice
that the projection of you on to V is going to be a scalar multiple
of vector V. The dot product would be a numerical value
and so would the magnitude of vector V squared.
Let's go ahead and use this formula to find some vector projections
and then we'll take a look at it graphically as well.
First we'll take a look at an example in R 2.
We want to determine the projection of U on to V given these two vectors
and then determine the vector component of U that's orthogonal to V. So here's our formula
to determine the vector projection of you on to V. It's going to be equal to U dotted
with V divided by the magnitude of vector V squared times vector V. Let's go
and determine the dot products.
So U dotted with V is going to be 3 times 4 that's 12 plus 1 times 2 that's 2 divided
by the magnitude of vector V squared.
Remember the magnitude of vector V would be the square root of 4 squared plus 2 squared
but if we're squaring the square root we're just going to have 4 squared plus 2 squared.
And we're going to multiply this scalar by vector V which is the vector 4, 2.
Let's go ahead and determine what this scalar would be, 14 divided by 16 plus 4 that's 20.
This fraction simplifies.
It will be 7/10 so we multiply this.
We're going to have 28/10 that will be 14/5, 7/10 times 2 would be 14/10 or 7/5.
And we'll go ahead and let the projection of U on to V be vector W sub 1.
Now we want to determine the vector component of U that's orthogonal to V. We'll go ahead
and call that W 2 and that's going to be equal to U minus W sub 1.
So we're going to have vector 3, 1 minus the vector 14/5,
7/5 so vector W2 is going to equal 3 minus 14/5.
That will be 1/5 and the Y component will be 1 minus 7/5 or negative 2/5.
So again here is the projection of U on to V and then here's the vector component
of U that's orthogonal to V. Let's go ahead and take a look at the graph of this.
We're projecting the red vector on to the blue vector.
The projection of U on to V would be here.
The orthogonal component or vector W sub 2 would be here.
And we can see that W sub 1 plus W sub two is going to equal vector U,
so here's a nice illustration of what we just found.
The projection highlighted in yellow and the orthogonal component in green.
Let's go ahead and try one in R 3.
We want to find the projection of vector U on to vector V so go ahead
and call the projection vector W sub 1.
So the numerator will be U dotted with V
so we'll have 1 times 5 that's 5 plus 2 times 4 that's 8 plus 1 times 2 that's 2 divided
by the magnitude of vector V squared.
That will just give us 5 squared plus 4 squared plus 2 squared.
We're going to multiply this by vector V which has components 5, 4 and 2.
Let's go ahead and simplify this.
We're going to have 5 plus 8 plus 2 that's 15 divided
by 25 plus 16 that's 41 plus 4 that's 45.
This simplifies to 1/3 so this is going to give us 5/3, 4/3, and 2/3.
Again this is the projection of U on to V which we're calling W sub 1.
So now to determine the vector component of U orthogonal
to V we'll call it W sub 2 it's going to be equal to U minus W sub 1.
So we're going to have 1, 2, 1 minus 5/3, 4/3, and 2/3.
So we'll have 1 minus 5/3.
That's going to be negative 2/3.
We'll have 2 minus 4/3.
That will be 2/3, and 1 minus 2/3 is 1/3 so again here's our projection of U on to V.
And here's our vector component of U that is orthogonal
to V. Let's look at this one graphically as well.
It's a little bit harder to interpret in 3D but if you project the red vector
on to the blue vector the projection would be this gray vector here
which we called vector W sub 1 and then the vector component of U that's orthogonal
to V would be this green vector here which we called vector W sub 2.
Okay, that's going to do it for this video.
I hope you found these examples helpful.
Thank you for watching. ------------------------------3f6219d73d12--