JOEL LEWIS: Hi.
Welcome back to recitation.
In lecture, among other things, you've been learning
about computing components of one vector in the direction of
another vector.
So I have a straightforward problem about
that for you here.
So we've got two vectors.
The vector 2i minus 2j plus k.
And we've got the vector i plus j plus k.
And so what I'd like you to do for me is to compute the
component of this first vector in the direction
of the second vector.
So why don't you pause the video, take some time to work
that out, come back, and we can work on it together.
Welcome back.
So hopefully you had some luck with this problem.
Now this problem is pretty straightforward.
Really all you have to do is remember what the definition
of component is.
And after that, it's smooth sailing.
So in particular, the component of one vector in the
direction of another is the length of what you get when
you project this vector onto that one, well,
plus a sign, right?
So if the projection is in the same direction, then it's
positive, or if it's in the opposite
direction, it's negative.
So let me draw a picture of what I mean by that.
So if you have a vector V and you have another vector W,
then the projection of V onto W is what you get if you make
a, drop a perpendicular line there, and then it's just this
vector here.
So that's the projection.
And then the component is the length of that projection.
Again, with the sign if necessary.
And so we can see since this is going to be a right
triangle here, we can see that this vector has length that's
just given by the length of V. So the length-- rather-- the
component of V in direction W is, so it's just the
length of V, right.
That's the length of the hypotenuse--
times the cosine of the angle between them--
so it's times cosine of theta.
Now this length of V times cosine theta, this should
remind you of something.
This looks very much like this formula we had for the dot
product, right?
So another way of writing this is to say that this is equal
to V dot W.
So V dot W is the length of V times the length of W times
the cosine of the angle.
And so here we just have the length of V times the cosine
of the angle.
So we have to divide this through by the length of W. So
this is what the component is, and this is the simple formula
for it if you're given V and W in some easy-to-use form.
And indeed in this problem, we're given V and W just in
they're nice coordinate forms.
So we're given that our vector V that we want the component
of is 2i minus 2j plus k.
And the direction W that we're looking at is i plus j plus k.
So in our case, we just have to compute these expressions V
dot W and the length of W in order to put them into this
formula, and then we'll be done.
So in our case, V dot W-- well, that's straightforward
because we're given V and W in coordinates--
so this is just 2 times 1 plus minus 2 times 1
plus 1 times 1.
So that's, OK, 2 minus 2 plus 1 is 1.
And the length of W-- well, you know, it's just your usual
length formula--
is 1 squared plus 1 squared plus 1 squared, the whole
thing is square rooted, which is equal to the
square root of 3.
So we've got V dot W and we have the length of W, and so
then we just need to put them right into this formula.
That the component of V in the direction of W is given by
this expression.
So the component of--
I'm not going to write it out with i's, j's, and k's, I'm
going to write--
2, minus 2, 1.
And direction 1, 1, 1 is, well we just have to divide the dot
product by the length of the direction vector.
So that's 1 divided by the square root of 3.
So that's that, I'll end there.