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PxCxlsl_YwY
using this dot product, so, by the way,
PxCxlsl_YwY
I should point out, we put this dot here.
PxCxlsl_YwY
That's why it's called dot product.
PxCxlsl_YwY
So, what this tells us is we should get the same thing as
PxCxlsl_YwY
multiplying the length of A with itself, so, squared,
PxCxlsl_YwY
times the cosine of the angle. But now, the cosine of an
PxCxlsl_YwY
angle, of zero, cosine of zero you all know is
PxCxlsl_YwY
one. OK, so that's going to be
PxCxlsl_YwY
length A^2. Well, doesn't stand a chance of
PxCxlsl_YwY
being true? Well, let's see.
PxCxlsl_YwY
If we do AdotA using this formula, we will get a1^2 a2^2
PxCxlsl_YwY
a3^2. That is, indeed,
PxCxlsl_YwY
the square of the length. So, check.
PxCxlsl_YwY
That works. OK, now, what about two
PxCxlsl_YwY
different vectors? Can we understand what this
PxCxlsl_YwY
says, and how it relates to that?
PxCxlsl_YwY
So, let's say that I have two different vectors,
PxCxlsl_YwY
A and B, and I want to try to understand what's going on.
PxCxlsl_YwY
So, my claim is that we are going to be able to understand
PxCxlsl_YwY
the relation between this and that in terms of the law of
PxCxlsl_YwY
cosines. So, the law of cosines is
PxCxlsl_YwY
something that tells you about the length of the third side in
PxCxlsl_YwY
the triangle like this in terms of these two sides,
PxCxlsl_YwY
and the angle here. OK, so the law of cosines,
PxCxlsl_YwY
which hopefully you have seen before, says that,
PxCxlsl_YwY
so let me give a name to this side.
PxCxlsl_YwY
Let's call this side C, and as a vector,
PxCxlsl_YwY
C is A minus B. It's minus B plus A.
PxCxlsl_YwY
So, it's getting a bit cluttered here.
PxCxlsl_YwY
So, the law of cosines says that the length of the third
PxCxlsl_YwY
side in this triangle is equal to length A2 plus length B2.
PxCxlsl_YwY
Well, if I stopped here, that would be Pythagoras,
PxCxlsl_YwY
but I don't have a right angle. So, I have a third term which
PxCxlsl_YwY
is twice length A, length B, cosine theta,
PxCxlsl_YwY
OK? Has everyone seen this formula
PxCxlsl_YwY
sometime? I hear some yeah's.
PxCxlsl_YwY
I hear some no's. Well, it's a fact about,
PxCxlsl_YwY
I mean, you probably haven't seen it with vectors,
PxCxlsl_YwY
but it's a fact about the side lengths in a triangle.
PxCxlsl_YwY
And, well, let's say, if you haven't seen it before,
PxCxlsl_YwY
then this is going to be a proof of the law of cosines if
PxCxlsl_YwY
you believe this. Otherwise, it's the other way
PxCxlsl_YwY
around. So, let's try to see how this
PxCxlsl_YwY
relates to what I'm saying about the dot product.
PxCxlsl_YwY
So, I've been saying that length C^2, that's the same
PxCxlsl_YwY
thing as CdotC, OK?
PxCxlsl_YwY
That, we have checked. Now, CdotC, well,
PxCxlsl_YwY
C is A minus B. So, it's A minus B,
PxCxlsl_YwY
dot product, A minus B.
PxCxlsl_YwY
Now, what do we want to do in a situation like that?
PxCxlsl_YwY
Well, we want to expand this into a sum of four terms.
PxCxlsl_YwY
Are we allowed to do that? Well, we have this dot product
PxCxlsl_YwY
that's a mysterious new operation.
PxCxlsl_YwY
We don't really know. Well, the answer is yes,
PxCxlsl_YwY
we can do it. You can check from this
PxCxlsl_YwY
definition that it behaves in the usual way in terms of
PxCxlsl_YwY
expanding, vectoring, and so on.
PxCxlsl_YwY
So, I can write that as AdotA minus AdotB minus BdotA plus
PxCxlsl_YwY
BdotB. So, AdotA is length A^2.
PxCxlsl_YwY
Let me jump ahead to the last term.
PxCxlsl_YwY
BdotB is length B^2, and then these two terms,
PxCxlsl_YwY
well, they're the same. You can check from the
PxCxlsl_YwY
definition that AdotB and BdotA are the same thing.
PxCxlsl_YwY
Well, you see that this term, I mean, this is the only
PxCxlsl_YwY
difference between these two formulas for the length of C.
PxCxlsl_YwY
So, if you believe in the law of cosines, then it tells you
PxCxlsl_YwY
that, yes, this a proof that AdotB equals length A length B
PxCxlsl_YwY
cosine theta. Or, vice versa,
PxCxlsl_YwY
if you've never seen the law of cosines, you are willing to
PxCxlsl_YwY
believe this. Then, this is the proof of the
PxCxlsl_YwY
law of cosines. So, the law of cosines,
PxCxlsl_YwY
or this interpretation, are equivalent to each other.
PxCxlsl_YwY
OK, any questions? Yes?
PxCxlsl_YwY
So, in the second one there isn't a cosine theta because I'm
PxCxlsl_YwY
just expanding a dot product. OK, so I'm just writing C
PxCxlsl_YwY
equals A minus B, and then I'm expanding this
PxCxlsl_YwY
algebraically. And then, I get to an answer
PxCxlsl_YwY
that has an A.B. So then, if I wanted to express
PxCxlsl_YwY
that without a dot product, then I would have to introduce
PxCxlsl_YwY
a cosine. And, I would get the same as
PxCxlsl_YwY
that, OK? So, yeah, if you want,
PxCxlsl_YwY
the next step to recall the law of cosines would be plug in this
PxCxlsl_YwY
formula for AdotB. And then you would have a
PxCxlsl_YwY
cosine. OK, let's keep going.
PxCxlsl_YwY
OK, so what is this good for? Now that we have a definition,
PxCxlsl_YwY
we should figure out what we can do with it.
PxCxlsl_YwY
So, what are the applications of dot product?
PxCxlsl_YwY
Well, will this discover new applications of dot product
PxCxlsl_YwY
throughout the entire semester,but let me tell you at
PxCxlsl_YwY
least about those that are readily visible.
PxCxlsl_YwY
So, one is to compute lengths and angles, especially angles.
PxCxlsl_YwY
So, let's do an example. Let's say that,
PxCxlsl_YwY
for example, I have in space,
PxCxlsl_YwY
I have a point, P, which is at (1,0,0).
PxCxlsl_YwY
I have a point, Q, which is at (0,1,0).
PxCxlsl_YwY
So, it's at distance one here, one here.
PxCxlsl_YwY
And, I have a third point, R at (0,0,2),
PxCxlsl_YwY
so it's at height two. And, let's say that I'm
PxCxlsl_YwY
curious, and I'm wondering what is the angle here?
PxCxlsl_YwY
So, here I have a triangle in space connect P,