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JOEL LEWIS: Hi.

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Welcome back to recitation.

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In lecture, you've learned
a little bit about the dot

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product, and how
the dot product can

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be used to compute
angles between vectors,

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or the relationship between
the dot product of vectors

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and the angle between them.

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So, here I have two questions,
two related questions for you.

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So the first one,
you've got two vectors.

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i hat plus j hat plus 2 k hat.

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And a second vector, 2 i
hat minus j hat plus k hat.

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I'm probably going to stop
saying the hats in a minute,

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just warning you.

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So you've got these
two different vectors,

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and what I'd like to know is
what the angle between them is.

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And the second one is
that we have three points.

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So the P has coordinates
a, 1, and minus 1.

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Q has coordinates 0, 1, and 1.

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And R has coordinates
a, minus 1, and 3.

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So when you've
given three points,

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that define some angles.

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So what I'd like to know
is for which values of a

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is the angle PQR a right angle?

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So why don't you take some
time, work out these problems,

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come back, and we can
work them out together.

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Hopefully you had some
luck on these problems.

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Let's get started.

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Let's start with the first one.

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So we have these two vectors, i
plus j plus 2k, and 2i minus j

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plus k.

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And we want to know
what the measure

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of the angle between them is.

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So the thing that
we have to use here,

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is that for vectors
v and w, we know

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that v dot w can
be written in terms

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of the angle between them.

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So in particular, we know
that for vectors v and w,

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if the angle between
them is equal to theta,

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then cosine of theta
is equal to v dot w,

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but I have to divide
by the length of v

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and the length of w.

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So I have this simple
formula for cosine

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of theta in terms of v and w.

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And so then if we
want, we can find theta

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by taking an arccosine.

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So let's apply this formula
in the case of problem a.

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So we could say that v is
equal to i plus j plus 2k,

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and w is equal to
2i minus j plus k.

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And so what that means is that
in our case, v dot w-- well,

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we just take, you know,
the products of the

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coordinates and add them up.

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So in that case, this is 1
times 2 plus 1 times minus 1

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plus 2 times 1.

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So this is 1 times 2 plus 1
times minus 1 plus 2 times 1.

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So that's 2 minus 1 plus
2, so that's equal to 3.

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And so then we also need to
know what the length of v and w

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are in order to
apply this formula.

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So the length of v is
equal to-- well it's

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the square root of the sum of
the squares of the coordinates,

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or the square root of v dot
v. So in this case, that's

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the square root of 1 squared
plus 1 squared plus 2 squared.

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So that's the square root of 6.

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And the length of w
is equal to-- it's

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the square root of 2
squared plus minus 1

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squared plus 1
squared, which is also

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equal to the square root of 6.

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So if you put these
three things together,

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we have that cosine
of theta is equal--

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so I'm going to write
it right up here

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next to this
general formula just

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to save a little bit of
space-- so in our case,

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this is equal to 3 divided
by the square root of 6

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times the square root of 6.

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So that's 3 divided by
6, which is equal to 1/2.

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So if cosine of theta
is equal to 1/2,

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well we know that
that means that theta

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is equal to pi over 3.

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OK?

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So the angle between these
two vectors is pi over 3.

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OK, great.

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That's part a.

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Now let's talk about part b.

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So we have these
three points and we

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want to know for what values
is the angle that they

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define a right angle.

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For what values
of this parameter.

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Right?

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Our points depend on
this parameter, a.

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So in order to do that, we have
these three points in space.

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Let me draw a picture up here.

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So we have point P somewhere
and point Q somewhere and point

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R somewhere, and they
define this angle,

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and we want to know when
this angle is a right angle.

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Well, we don't have
any vectors yet,

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but it's easy to get
some vectors involved

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in this problem.

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We can look at the
vectors QP and QR.

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So those two vectors
are the two sides

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of this angle that
we're interested in.

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And it's easy when you're given
coordinates for points to find

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the vector between them.

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So you just subtract, right?

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So QP.

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In order to get from
our point Q over here--

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in order to get from (0,
1, 1) to (a, 1, -1)-- well,

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you have to increase the first
coordinate by a, the second

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coordinate has to stay the same,
and the third coordinate has

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to decrease by 2.

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So this vector QP is
equal to [a, 0, -2].

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So now I'm using angle brackets,
because this is a vector.

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SPEAKER 1: Positive
2 or negative 2?

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JOEL LEWIS: Negative 2.

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Yes.

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Right.

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Negative 2, yeah?

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Because we're going
from 1 to minus 1.

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OK, good.

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And similarly QR.

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So to get from Q to R-- I
have to go back and look

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at the coordinates
over here-- so I

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increase the first
coordinate by a,

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decrease the second
coordinate by 2,

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increase the third coordinate
by 2, to get from Q to R.

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So the vector QR is equal to
a-- what did I say-- minus 2, 2.

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So these are the
vectors QP and QR,

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and we want to know when the
angle between these two things

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is a right angle.

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We want to know when these
vectors are perpendicular.

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OK?

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So the two vectors are
perpendicular exactly when

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their dot product is 0.

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Right?

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When the angle is 90 degrees.

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Cosine of 90 degrees,
cosine of pi over 2 radians,

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is equal to 0, so those
two things are the same.

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The dot product equals 0
and the angle is 90 degrees.

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So OK.

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So in our case, we just have
to look then at QP dot QR.

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So that's equal to-- well,
we just take this dot product

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here-- so it's equal to a
squared plus-- 0 times minus 2

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is 0-- minus-- 2 times 2,
so plus minus 2 times 2-- so

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that's minus 4.

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So that's a squared minus 4.

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And so this dot
product is equal to 0

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exactly when a squared
minus 4 is equal to 0.

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So a squared minus 4 is equal
to 0 when a is equal to 2

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or when a is equal to minus 2.

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So there are two
values of a for which

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those three points form
a right angle at Q.

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So just to quickly summarize, we
used this general formula here

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for the cosine of an angle
in terms of the two vectors

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that are the sides of the angle.

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And with that formula, we
can compute the measure

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of angles between
two vectors as here--

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or, as in the first
part, in part a.

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Or in a special case when we're
looking for a 90 degree angle,

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we can use the simplified
form when we just

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need to figure out when the
dot product is equal to 0.

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OK.

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So I'll end there.