Welcome to a video that will prove the vector projection formula.
Let's start by reviewing vector projection.
So we want to project the red vector u onto the blue vector v.
We use this notation here.
And the result would be this gray vector here.
And the formula to determine this vector projection.
Would be u dotted with v
divided by the magnitude of v squared
times vector v.
So in this video we want to show
where this formula comes from.
So let's go ahead and set this up.
Again we'll call this vector u.
We'll call the blue vector
vector v.
So were projecting u under v.
And let's go ahead and call the projection
vector w.
So vector u and vector w
would form a right triangle.
And we'll call this angle between them
theta.
So using
right triangle trigonometry.
We can say cosine theta
would be equal to
the magnitude of w divided by the magnitude of u.
Remember our ultimate goal is to determine
the formula for
vector w.
Which is a projection of u onto v.
And let's go ahead and state that.
So if we wanted to solve this for the magnitude of w we were just multiply both sides by the magnitude of u.
So we have the magnitude of w.
Is equal to the magnitude of u.
Cosine theta.
Let's go ahead and call this equation one.
From an earlier proof.
We showed that cosine theta.
Is equal to u dotted with v.
Divided by the magnitude of u.
Times the magnitude of v.
And of course this gives us another way to determine the dot product for u and v.
We can say that u dotted with v.
Is equal to
the magnitude of u
times the magnitude of v times cosine theta.
Let's go ahead and call this equation two.
Now were gonna multiply.
Equation one by
the magnitude of v
over the magnitude of v which of course is just equal to one.
So were gonna take this equation here.
And now it's going to be
the magnitude of w.
It's gonna be equal to
the magnitude of u.
Times the magnitude of v
cosine theta
all over
the magnitude of v.
So again here's the magnitude of v divided by the magnitude of v that we multiplied by.
We call this equation three.
Now what were going to do is to perform a substitution
with equation two. Meaning we're gonna take this numerator here
and replace it with u
dotted with v.
Let's go ahead and do that on the next screen.
Let's go back and take a look at that again.
We took the numerator of this fraction here
and replaced it with
equation two
where the right side was equal to u dotted with v.
Now that we have a formula for the magnitude of vector w.
The last thing we need to do is make sure that its parallel to vector v.
Remember were projecting u onto v.
So if it's pointing in the same direction or at least parallel to vector v.
That must be the scalar multiple of v.
So were gonna take the magnitude of w.
And multiply it by the unit vector in the direction of v.
So vector w is gonna be equal to
it's magnitude which we found from this formula here.
times vector v normalized.
So we'll have vector v.
Divided by its magnitude.
And now we have the formula for the projection of u under v.
Which is vector w. So we have vector w is equal to
u dotted with v.
Divided by the magnitude of v squared.
Times vector v.
So we have the projection of u
onto v
to what we're trying to prove.
And there we have it.
I hope you found this prove helpful.
Thank you for watching.