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So let's start right away with
stuff that we will need to see
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before we can go on to more
advanced things.
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So, hopefully yesterday in
recitation, you heard a bit
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about vectors.
How many of you actually knew
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about vectors before that?
OK, that's the vast majority.
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If you are not one of those
people, well,
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hopefully you'll learn about
vectors right now.
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I'm sorry that the learning
curve will be a bit steeper for
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the first week.
But hopefully,
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you'll adjust fine.
If you have trouble with
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vectors, do go to your
recitation instructor's office
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hours for extra practice if you
feel the need to.
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You will see it's pretty easy.
So, just to remind you,
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a vector is a quantity that has
both a direction and a magnitude
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of length.
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So -- So, concretely the way
you draw a vector is by some
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arrow, like that,
OK?
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And so, it has a length,
and it's pointing in some
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direction.
And, so, now,
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the way that we compute things
with vectors,
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typically, as we introduce a
coordinate system.
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So, if we are in the plane,
x-y-axis, if we are in space,
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x-y-z axis.
So, usually I will try to draw
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my x-y-z axis consistently to
look like this.
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And then, I can represent my
vector in terms of its
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components along the coordinate
axis.
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So, that means when I have this
row, I can ask,
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how much does it go in the x
direction?
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00:02:15 --> 00:02:17
How much does it go in the y
direction?
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00:02:17 --> 00:02:20
How much does it go in the z
direction?
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00:02:20 --> 00:02:25
And, so, let's call this a
vector A.
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So, it's more convention.
When we have a vector quantity,
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we put an arrow on top to
remind us that it's a vector.
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If it's in the textbook,
then sometimes it's in bold
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because it's easier to typeset.
If you've tried in your
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favorite word processor,
bold is easy and vectors are
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not easy.
So, the vector you can try to
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decompose terms of unit vectors
directed along the coordinate
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axis.
So, the convention is there is
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a vector that we call
***amp***lt;i***amp***gt;
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hat that points along the x
axis and has length one.
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There's a vector called
***amp***lt;j***amp***gt;
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hat that does the same along
the y axis,
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and the
***amp***lt;k***amp***gt;
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hat that does the same along
the z axis.
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And, so, we can express any
vector in terms of its
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components.
So, the other notation is
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***amp***lt;a1,
a2, a3 ***amp***gt;
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between these square brackets.
Well, in angular brackets.
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So, the length of a vector we
denote by, if you want,
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it's the same notation as the
absolute value.
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So, that's going to be a
number, as we say,
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now, a scalar quantity.
OK, so, a scalar quantity is a
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usual numerical quantity as
opposed to a vector quantity.
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And, its direction is sometimes
called dir A,
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and that can be obtained just
by scaling the vector down to
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unit length,
for example,
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by dividing it by its length.
So -- Well, there's a lot of
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notation to be learned.
So, for example,
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if I have two points,
P and Q, then I can draw a
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vector from P to Q.
And, that vector is called
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vector PQ, OK?
So, maybe we'll call it A.
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But, a vector doesn't really
have, necessarily,
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a starting point and an ending
point.
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OK, so if I decide to start
here and I go by the same
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distance in the same direction,
this is also vector A.
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It's the same thing.
So, a lot of vectors we'll draw
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starting at the origin,
but we don't have to.
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So, let's just check and see
how things went in recitation.
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So, let's say that I give you
the vector
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***amp***lt;3,2,1***amp***gt;.
And so, what do you think about
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the length of this vector?
OK, I see an answer forming.
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So, a lot of you are answering
the same thing.
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Maybe it shouldn't spoil it for
those who haven't given it yet.
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OK, I think the overwhelming
vote is in favor of answer
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number two.
I see some sixes, I don't know.
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That's a perfectly good answer,
too, but hopefully in a few
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minutes it won't be I don't know
anymore.
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00:06:10 --> 00:06:17
So, let's see.
How do we find -- -- the length
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of a vector three,
two, one?
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00:06:24 --> 00:06:30
Well, so, this vector,
A, it comes towards us along
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the x axis by three units.
It goes to the right along the
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y axis by two units,
and then it goes up by one unit
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00:06:42 --> 00:06:46
along the z axis.
OK, so, it's pointing towards
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here.
That's pretty hard to draw.
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So, how do we get its length?
Well, maybe we can start with
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something easier,
the length of the vector in the
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plane.
So, observe that A is obtained
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from a vector,
B, in the plane.
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Say, B equals three (i) hat
plus two (j) hat.
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And then, we just have to,
still, go up by one unit,
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OK?
So, let me try to draw a
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picture in this vertical plane
that contains A and B.
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00:07:20 --> 00:07:23
If I draw it in the vertical
plane,
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so, that's the Z axis,
that's not any particular axis,
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then my vector B will go here,
and my vector A will go above
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it.
And here, that's one unit.
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And, here I have a right angle.
So, I can use the Pythagorean
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theorem to find that length A^2
equals length B^2 plus one.
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Now, we are reduced to finding
the length of B.
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The length of B,
we can again find using the
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Pythagorean theorem in the XY
plane because here we have the
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right angle.
Here we have three units,
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and here we have two units.
OK, so, if you do the
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calculations,
you will see that,
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well, length of B is square
root of (3^2 2^2),
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that's 13.
So, the square root of 13 -- --
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and length of A is square root
of length B^2 plus one (square
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00:08:32 --> 00:08:41
it if you want) which is going
to be square root of 13 plus one
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is the square root of 14,
hence, answer number two which
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almost all of you gave.
OK, so the general formula,
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if you follow it with it,
in general if we have a vector
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with components a1,
a2, a3,
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00:09:07 --> 00:09:16
then the length of A is the
square root of a1^2 plus a2^2
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plus a3^2.
OK, any questions about that?
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Yes?
Yes.
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00:09:29 --> 00:09:32
So, in general,
we indeed can consider vectors
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00:09:32 --> 00:09:36
in abstract spaces that have any
number of coordinates.
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And that you have more
components.
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In this class,
we'll mostly see vectors with
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two or three components because
they are easier to draw,
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and because a lot of the math
that we'll see works exactly the
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same way whether you have three
variables or a million
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variables.
If we had a factor with more
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components, then we would have a
lot of trouble drawing it.
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00:09:55 --> 00:09:58
But we could still define its
length in the same way,
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00:09:58 --> 00:10:01
by summing the squares of the
components.
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00:10:01 --> 00:10:04
So, I'm sorry to say that here,
multi-variable,
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00:10:04 --> 00:10:07
multi will mean mostly two or
three.
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00:10:07 --> 00:10:13
But, be assured that it works
just the same way if you have
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10,000 variables.
Just, calculations are longer.
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00:10:20 --> 00:10:28
OK, more questions?
So, what else can we do with
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00:10:28 --> 00:10:31
vectors?
Well, another thing that I'm
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00:10:31 --> 00:10:35
sure you know how to do with
vectors is to add them to scale
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00:10:35 --> 00:10:39
them.
So, vector addition,
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00:10:39 --> 00:10:48
so, if you have two vectors,
A and B, then you can form,
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00:10:48 --> 00:10:52
their sum, A plus B.
How do we do that?
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00:10:52 --> 00:10:54
Well, first,
I should tell you,
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00:10:54 --> 00:10:56
vectors, they have this double
life.
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00:10:56 --> 00:10:59
They are, at the same time,
geometric objects that we can
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00:10:59 --> 00:11:02
draw like this in pictures,
and there are also
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00:11:02 --> 00:11:06
computational objects that we
can represent by numbers.
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00:11:06 --> 00:11:09
So, every question about
vectors will have two answers,
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00:11:09 --> 00:11:11
one geometric,
and one numerical.
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00:11:11 --> 00:11:14
OK, so let's start with the
geometric.
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00:11:14 --> 00:11:17
So, let's say that I have two
vectors, A and B,
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00:11:17 --> 00:11:21
given to me.
And, let's say that I thought
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00:11:21 --> 00:11:24
of drawing them at the same
place to start with.
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00:11:24 --> 00:11:28
Well, to take the sum,
what I should do is actually
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00:11:28 --> 00:11:33
move B so that it starts at the
end of A, at the head of A.
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00:11:33 --> 00:11:38
OK, so this is, again, vector B.
So, observe,
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this actually forms,
now, a parallelogram,
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00:11:41 --> 00:11:43
right?
So, this side is,
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00:11:43 --> 00:11:48
again, vector A.
And now, if we take the
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diagonal of that parallelogram,
this is what we call A plus B,
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00:11:57 --> 00:12:00
OK, so, the idea being that to
move along A plus B,
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00:12:00 --> 00:12:03
it's the same as to move first
along A and then along B,
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00:12:03 --> 00:12:09
or, along B, then along A.
A plus B equals B plus A.
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00:12:09 --> 00:12:13
OK, now, if we do it
numerically,
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00:12:13 --> 00:12:19
then all you do is you just add
the first component of A with
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00:12:19 --> 00:12:23
the first component of B,
the second with the second,
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00:12:23 --> 00:12:28
and the third with the third.
OK, say that A was
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***amp***lt;a1,
a2, a3***amp***gt;
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00:12:31 --> 00:12:35
B was ***amp***lt;b1,
b2, b3***amp***gt;,
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00:12:35 --> 00:12:40
then you just add this way.
OK, so it's pretty
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straightforward.
So, for example,
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I said that my vector over
there, its components are three,
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00:12:48 --> 00:12:54
two, one.
But, I also wrote it as 3i 2j k.
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00:12:54 --> 00:12:57
What does that mean?
OK, so I need to tell you first
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00:12:57 --> 00:13:06
about multiplying by a scalar.
So, this is about addition.
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00:13:06 --> 00:13:11
So, multiplication by a scalar,
it's very easy.
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00:13:11 --> 00:13:15
If you have a vector,
A, then you can form a vector
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00:13:15 --> 00:13:20
2A just by making it go twice as
far in the same direction.
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00:13:20 --> 00:13:24
Or, we can make half A more
modestly.
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00:13:24 --> 00:13:31
We can even make minus A,
and so on.
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00:13:31 --> 00:13:35
So now, you see,
if I do the calculation,
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00:13:35 --> 00:13:38
3i 2j k, well,
what does it mean?
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00:13:38 --> 00:13:43
3i is just going to go along
the x axis, but by distance of
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00:13:43 --> 00:13:47
three instead of one.
And then, 2j goes two units
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00:13:47 --> 00:13:51
along the y axis,
and k goes up by one unit.
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00:13:51 --> 00:13:54
Well, if you add these
together, you will go from the
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00:13:54 --> 00:13:58
origin, then along the x axis,
then parallel to the y axis,
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00:13:58 --> 00:14:02
and then up.
And, you will end up,
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00:14:02 --> 00:14:05
indeed, at the endpoint of a
vector.
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00:14:05 --> 00:14:19
OK, any questions at this point?
Yes?
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00:14:19 --> 00:14:21
Exactly.
To add vectors geometrically,
194
00:14:21 --> 00:14:25
you just put the head of the
first vector and the tail of the
195
00:14:25 --> 00:14:30
second vector in the same place.
And then, it's head to tail
196
00:14:30 --> 00:14:35
addition.
Any other questions?
197
00:14:35 --> 00:14:41
Yes?
That's correct.
198
00:14:41 --> 00:14:43
If you subtract two vectors,
that just means you add the
199
00:14:43 --> 00:14:45
opposite of a vector.
So, for example,
200
00:14:45 --> 00:14:49
if I wanted to do A minus B,
I would first go along A and
201
00:14:49 --> 00:14:52
then along minus B,
which would take me somewhere
202
00:14:52 --> 00:14:55
over there, OK?
So, A minus B,
203
00:14:55 --> 00:15:01
if you want,
would go from here to here.
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00:15:01 --> 00:15:08
OK, so hopefully you've kind of
seen that stuff either before in
205
00:15:08 --> 00:15:13
your lives, or at least
yesterday.
206
00:15:13 --> 00:15:23
So, I'm going to use that as an
excuse to move quickly forward.
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00:15:23 --> 00:15:28
So, now we are going to learn a
few more operations about
208
00:15:28 --> 00:15:31
vectors.
And, these operations will be
209
00:15:31 --> 00:15:34
useful to us when we start
trying to do a bit of geometry.
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00:15:34 --> 00:15:37
So, of course,
you've all done some geometry.
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00:15:37 --> 00:15:40
But, we are going to see that
geometry can be done using
212
00:15:40 --> 00:15:42
vectors.
And, in many ways,
213
00:15:42 --> 00:15:44
it's the right language for
that,
214
00:15:44 --> 00:15:47
and in particular when we learn
about functions we really will
215
00:15:47 --> 00:15:51
want to use vectors more than,
maybe, the other kind of
216
00:15:51 --> 00:15:54
geometry that you've seen
before.
217
00:15:54 --> 00:15:56
I mean, of course,
it's just a language in a way.
218
00:15:56 --> 00:15:59
I mean, we are just
reformulating things that you
219
00:15:59 --> 00:16:02
have seen, you already know
since childhood.
220
00:16:02 --> 00:16:07
But, you will see that notation
somehow helps to make it more
221
00:16:07 --> 00:16:10
straightforward.
So, what is dot product?
222
00:16:10 --> 00:16:16
Well, dot product as a way of
multiplying two vectors to get a
223
00:16:16 --> 00:16:21
number, a scalar.
And, well, let me start by
224
00:16:21 --> 00:16:25
giving you a definition in terms
of components.
225
00:16:25 --> 00:16:29
What we do, let's say that we
have a vector,
226
00:16:29 --> 00:16:32
A, with components a1,
a2, a3, vector B with
227
00:16:32 --> 00:16:34
components b1,
b2, b3.
228
00:16:34 --> 00:16:38
Well, we multiply the first
components by the first
229
00:16:38 --> 00:16:43
components, the second by the
second, the third by the third.
230
00:16:43 --> 00:16:46
If you have N components,
you keep going.
231
00:16:46 --> 00:16:49
And, you sum all of these
together.
232
00:16:49 --> 00:16:55
OK, and important:
this is a scalar.
233
00:16:55 --> 00:16:59
OK, you do not get a vector.
You get a number.
234
00:16:59 --> 00:17:01
I know it sounds completely
obvious from the definition
235
00:17:01 --> 00:17:03
here,
but in the middle of the action
236
00:17:03 --> 00:17:07
when you're going to do
complicated problems,
237
00:17:07 --> 00:17:14
it's sometimes easy to forget.
So, that's the definition.
238
00:17:14 --> 00:17:17
What is it good for?
Why would we ever want to do
239
00:17:17 --> 00:17:20
that?
That's kind of a strange
240
00:17:20 --> 00:17:23
operation.
So, probably to see what it's
241
00:17:23 --> 00:17:27
good for, I should first tell
you what it is geometrically.
242
00:17:27 --> 00:17:29
OK, so what does it do
geometrically?
243
00:17:29 --> 00:17:38
244
00:17:38 --> 00:17:42
Well, what you do when you
multiply two vectors in this
245
00:17:42 --> 00:17:45
way,
I claim the answer is equal to
246
00:17:45 --> 00:17:51
the length of A times the length
of B times the cosine of the
247
00:17:51 --> 00:17:59
angle between them.
So, I have my vector, A,
248
00:17:59 --> 00:18:04
and if I have my vector, B,
and I have some angle between
249
00:18:04 --> 00:18:06
them,
I multiply the length of A
250
00:18:06 --> 00:18:10
times the length of B times the
cosine of that angle.
251
00:18:10 --> 00:18:13
So, that looks like a very
artificial operation.
252
00:18:13 --> 00:18:16
I mean, why would want to do
that complicated multiplication?
253
00:18:16 --> 00:18:21
Well, the basic answer is it
tells us at the same time about
254
00:18:21 --> 00:18:25
lengths and about angles.
And, the extra bonus thing is
255
00:18:25 --> 00:18:29
that it's very easy to compute
if you have components,
256
00:18:29 --> 00:18:32
see, that formula is actually
pretty easy.
257
00:18:32 --> 00:18:39
So, OK, maybe I should first
tell you, how do we get this
258
00:18:39 --> 00:18:41
from that?
Because, you know,
259
00:18:41 --> 00:18:44
in math, one tries to justify
everything to prove theorems.
260
00:18:44 --> 00:18:45
So, if you want,
that's the theorem.
261
00:18:45 --> 00:18:47
That's the first theorem in
18.02.
262
00:18:47 --> 00:18:52
So, how do we prove the theorem?
How do we check that this is,
263
00:18:52 --> 00:18:55
indeed, correct using this
definition?
264
00:18:55 --> 00:19:06
So, in more common language,
what does this geometric
265
00:19:06 --> 00:19:11
definition mean?
Well, the first thing it means,
266
00:19:11 --> 00:19:14
before we multiply two vectors,
let's start multiplying a
267
00:19:14 --> 00:19:17
vector with itself.
That's probably easier.
268
00:19:17 --> 00:19:19
So, if we multiply a vector,
A, with itself,
269
00:19:19 --> 00:19:22
using this dot product,
so, by the way,
270
00:19:22 --> 00:19:24
I should point out,
we put this dot here.
271
00:19:24 --> 00:19:28
That's why it's called dot
product.
272
00:19:28 --> 00:19:33
So, what this tells us is we
should get the same thing as
273
00:19:33 --> 00:19:38
multiplying the length of A with
itself, so, squared,
274
00:19:38 --> 00:19:43
times the cosine of the angle.
But now, the cosine of an
275
00:19:43 --> 00:19:49
angle, of zero,
cosine of zero you all know is
276
00:19:49 --> 00:19:52
one.
OK, so that's going to be
277
00:19:52 --> 00:19:56
length A^2.
Well, doesn't stand a chance of
278
00:19:56 --> 00:19:57
being true?
Well, let's see.
279
00:19:57 --> 00:20:03
If we do AdotA using this
formula, we will get a1^2 a2^2
280
00:20:03 --> 00:20:07
a3^2.
That is, indeed,
281
00:20:07 --> 00:20:14
the square of the length.
So, check.
282
00:20:14 --> 00:20:18
That works.
OK, now, what about two
283
00:20:18 --> 00:20:23
different vectors?
Can we understand what this
284
00:20:23 --> 00:20:27
says, and how it relates to
that?
285
00:20:27 --> 00:20:33
So, let's say that I have two
different vectors,
286
00:20:33 --> 00:20:40
A and B, and I want to try to
understand what's going on.
287
00:20:40 --> 00:20:45
So, my claim is that we are
going to be able to understand
288
00:20:45 --> 00:20:49
the relation between this and
that in terms of the law of
289
00:20:49 --> 00:20:52
cosines.
So, the law of cosines is
290
00:20:52 --> 00:20:56
something that tells you about
the length of the third side in
291
00:20:56 --> 00:21:00
the triangle like this in terms
of these two sides,
292
00:21:00 --> 00:21:07
and the angle here.
OK, so the law of cosines,
293
00:21:07 --> 00:21:11
which hopefully you have seen
before, says that,
294
00:21:11 --> 00:21:14
so let me give a name to this
side.
295
00:21:14 --> 00:21:19
Let's call this side C,
and as a vector,
296
00:21:19 --> 00:21:29
C is A minus B.
It's minus B plus A.
297
00:21:29 --> 00:21:37
So, it's getting a bit
cluttered here.
298
00:21:37 --> 00:21:45
So, the law of cosines says
that the length of the third
299
00:21:45 --> 00:21:53
side in this triangle is equal
to length A2 plus length B2.
300
00:21:53 --> 00:21:56
Well, if I stopped here,
that would be Pythagoras,
301
00:21:56 --> 00:22:01
but I don't have a right angle.
So, I have a third term which
302
00:22:01 --> 00:22:07
is twice length A,
length B, cosine theta,
303
00:22:07 --> 00:22:10
OK?
Has everyone seen this formula
304
00:22:10 --> 00:22:13
sometime?
I hear some yeah's.
305
00:22:13 --> 00:22:16
I hear some no's.
Well, it's a fact about,
306
00:22:16 --> 00:22:19
I mean, you probably haven't
seen it with vectors,
307
00:22:19 --> 00:22:22
but it's a fact about the side
lengths in a triangle.
308
00:22:22 --> 00:22:27
And, well, let's say,
if you haven't seen it before,
309
00:22:27 --> 00:22:32
then this is going to be a
proof of the law of cosines if
310
00:22:32 --> 00:22:39
you believe this.
Otherwise, it's the other way
311
00:22:39 --> 00:22:43
around.
So, let's try to see how this
312
00:22:43 --> 00:22:47
relates to what I'm saying about
the dot product.
313
00:22:47 --> 00:22:54
So, I've been saying that
length C^2, that's the same
314
00:22:54 --> 00:22:56
thing as CdotC,
OK?
315
00:22:56 --> 00:23:01
That, we have checked.
Now, CdotC, well,
316
00:23:01 --> 00:23:06
C is A minus B.
So, it's A minus B,
317
00:23:06 --> 00:23:09
dot product,
A minus B.
318
00:23:09 --> 00:23:11
Now, what do we want to do in a
situation like that?
319
00:23:11 --> 00:23:16
Well, we want to expand this
into a sum of four terms.
320
00:23:16 --> 00:23:19
Are we allowed to do that?
Well, we have this dot product
321
00:23:19 --> 00:23:22
that's a mysterious new
operation.
322
00:23:22 --> 00:23:24
We don't really know.
Well, the answer is yes,
323
00:23:24 --> 00:23:27
we can do it.
You can check from this
324
00:23:27 --> 00:23:31
definition that it behaves in
the usual way in terms of
325
00:23:31 --> 00:23:34
expanding, vectoring,
and so on.
326
00:23:34 --> 00:23:49
So, I can write that as AdotA
minus AdotB minus BdotA plus
327
00:23:49 --> 00:23:55
BdotB.
So, AdotA is length A^2.
328
00:23:55 --> 00:23:56
Let me jump ahead to the last
term.
329
00:23:56 --> 00:24:01
BdotB is length B^2,
and then these two terms,
330
00:24:01 --> 00:24:04
well, they're the same.
You can check from the
331
00:24:04 --> 00:24:07
definition that AdotB and BdotA
are the same thing.
332
00:24:07 --> 00:24:20
333
00:24:20 --> 00:24:24
Well, you see that this term,
I mean, this is the only
334
00:24:24 --> 00:24:30
difference between these two
formulas for the length of C.
335
00:24:30 --> 00:24:34
So, if you believe in the law
of cosines, then it tells you
336
00:24:34 --> 00:24:39
that, yes, this a proof that
AdotB equals length A length B
337
00:24:39 --> 00:24:41
cosine theta.
Or, vice versa,
338
00:24:41 --> 00:24:45
if you've never seen the law of
cosines, you are willing to
339
00:24:45 --> 00:24:49
believe this.
Then, this is the proof of the
340
00:24:49 --> 00:24:53
law of cosines.
So, the law of cosines,
341
00:24:53 --> 00:24:59
or this interpretation,
are equivalent to each other.
342
00:24:59 --> 00:25:07
OK, any questions?
Yes?
343
00:25:07 --> 00:25:12
So, in the second one there
isn't a cosine theta because I'm
344
00:25:12 --> 00:25:16
just expanding a dot product.
OK, so I'm just writing C
345
00:25:16 --> 00:25:19
equals A minus B,
and then I'm expanding this
346
00:25:19 --> 00:25:22
algebraically.
And then, I get to an answer
347
00:25:22 --> 00:25:24
that has an A.B.
So then, if I wanted to express
348
00:25:24 --> 00:25:27
that without a dot product,
then I would have to introduce
349
00:25:27 --> 00:25:31
a cosine.
And, I would get the same as
350
00:25:31 --> 00:25:34
that, OK?
So, yeah, if you want,
351
00:25:34 --> 00:25:38
the next step to recall the law
of cosines would be plug in this
352
00:25:38 --> 00:25:43
formula for AdotB.
And then you would have a
353
00:25:43 --> 00:25:58
cosine.
OK, let's keep going.
354
00:25:58 --> 00:26:03
OK, so what is this good for?
Now that we have a definition,
355
00:26:03 --> 00:26:06
we should figure out what we
can do with it.
356
00:26:06 --> 00:26:11
So, what are the applications
of dot product?
357
00:26:11 --> 00:26:14
Well, will this discover new
applications of dot product
358
00:26:14 --> 00:26:17
throughout the entire
semester,but let me tell you at
359
00:26:17 --> 00:26:20
least about those that are
readily visible.
360
00:26:20 --> 00:26:33
So, one is to compute lengths
and angles, especially angles.
361
00:26:33 --> 00:26:39
So, let's do an example.
Let's say that,
362
00:26:39 --> 00:26:44
for example,
I have in space,
363
00:26:44 --> 00:26:51
I have a point,
P, which is at (1,0,0).
364
00:26:51 --> 00:26:55
I have a point,
Q, which is at (0,1,0).
365
00:26:55 --> 00:26:58
So, it's at distance one here,
one here.
366
00:26:58 --> 00:27:03
And, I have a third point,
R at (0,0,2),
367
00:27:03 --> 00:27:07
so it's at height two.
And, let's say that I'm
368
00:27:07 --> 00:27:11
curious, and I'm wondering what
is the angle here?
369
00:27:11 --> 00:27:15
So, here I have a triangle in
space connect P,
370
00:27:15 --> 00:27:20
Q, and R, and I'm wondering,
what is this angle here?
371
00:27:20 --> 00:27:23
OK, so, of course,
one solution is to build a
372
00:27:23 --> 00:27:25
model and then go and measure
the angle.
373
00:27:25 --> 00:27:28
But, we can do better than that.
We can just find the angle
374
00:27:28 --> 00:27:32
using dot product.
So, how would we do that?
375
00:27:32 --> 00:27:38
Well, so, if we look at this
formula, we see,
376
00:27:38 --> 00:27:44
so, let's say that we want to
find the angle here.
377
00:27:44 --> 00:27:50
Well, let's look at the formula
for PQdotPR.
378
00:27:50 --> 00:27:56
Well, we said it should be
length PQ times length PR times
379
00:27:56 --> 00:27:59
the cosine of the angle,
OK?
380
00:27:59 --> 00:28:01
Now, what do we know,
and what do we not know?
381
00:28:01 --> 00:28:04
Well, certainly at this point
we don't know the cosine of the
382
00:28:04 --> 00:28:06
angle.
That's what we would like to
383
00:28:06 --> 00:28:08
find.
The lengths,
384
00:28:08 --> 00:28:11
certainly we can compute.
We know how to find these
385
00:28:11 --> 00:28:14
lengths.
And, this dot product we know
386
00:28:14 --> 00:28:17
how to compute because we have
an easy formula here.
387
00:28:17 --> 00:28:20
OK, so we can compute
everything else and then find
388
00:28:20 --> 00:28:25
theta.
So, I'll tell you what we will
389
00:28:25 --> 00:28:31
do is we will find theta -- --
in this way.
390
00:28:31 --> 00:28:34
We'll take the dot product of
PQ with PR, and then we'll
391
00:28:34 --> 00:28:36
divide by the lengths.
392
00:28:36 --> 00:29:14
393
00:29:14 --> 00:29:27
OK, so let's see.
So, we said cosine theta is
394
00:29:27 --> 00:29:33
PQdotPR over length PQ length
PR.
395
00:29:33 --> 00:29:36
So, let's try to figure out
what this vector,
396
00:29:36 --> 00:29:39
PQ,
well, to go from P to Q,
397
00:29:39 --> 00:29:43
I should go minus one unit
along the x direction plus one
398
00:29:43 --> 00:29:46
unit along the y direction.
And, I'm not moving in the z
399
00:29:46 --> 00:29:49
direction.
So, to go from P to Q,
400
00:29:49 --> 00:29:54
I have to move by
***amp***lt;-1,1,0***amp***gt;.
401
00:29:54 --> 00:29:59
To go from P to R,
I go -1 along the x axis and 2
402
00:29:59 --> 00:30:04
along the z axis.
So, PR, I claim, is this.
403
00:30:04 --> 00:30:12
OK, then, the lengths of these
vectors, well,(-1)^2 (1)^2
404
00:30:12 --> 00:30:19
(0)^2, square root,
and then same thing with the
405
00:30:19 --> 00:30:24
other one.
OK, so, the denominator will
406
00:30:24 --> 00:30:30
become the square root of 2,
and there's a square root of 5.
407
00:30:30 --> 00:30:34
What about the numerator?
Well, so, remember,
408
00:30:34 --> 00:30:37
to do the dot product,
we multiply this by this,
409
00:30:37 --> 00:30:40
and that by that,
that by that.
410
00:30:40 --> 00:30:45
And, we add.
Minus 1 times minus 1 makes 1
411
00:30:45 --> 00:30:49
plus 1 times 0,
that's 0.
412
00:30:49 --> 00:30:55
Zero times 2 is 0 again.
So, we will get 1 over square
413
00:30:55 --> 00:30:59
root of 10.
That's the cosine of the angle.
414
00:30:59 --> 00:31:03
And, of course if we want the
actual angle,
415
00:31:03 --> 00:31:08
well, we have to take a
calculator, find the inverse
416
00:31:08 --> 00:31:12
cosine, and you'll find it's
about 71.5°.
417
00:31:12 --> 00:31:18
Actually, we'll be using mostly
radians, but for today,
418
00:31:18 --> 00:31:26
that's certainly more speaking.
OK, any questions about that?
419
00:31:26 --> 00:31:29
No?
OK, so in particular,
420
00:31:29 --> 00:31:32
I should point out one thing
that's really neat about the
421
00:31:32 --> 00:31:34
answer.
I mean, we got this number.
422
00:31:34 --> 00:31:37
We don't really know what it
means exactly because it mixes
423
00:31:37 --> 00:31:39
together the lengths and the
angle.
424
00:31:39 --> 00:31:41
But, one thing that's
interesting here,
425
00:31:41 --> 00:31:45
it's the sign of the answer,
the fact that we got a positive
426
00:31:45 --> 00:31:48
number.
So, if you think about it,
427
00:31:48 --> 00:31:50
the lengths are always
positive.
428
00:31:50 --> 00:31:56
So, the sign of a dot product
is the same as a sign of cosine
429
00:31:56 --> 00:32:00
theta.
So, in fact,
430
00:32:00 --> 00:32:13
the sign of AdotB is going to
be positive if the angle is less
431
00:32:13 --> 00:32:17
than 90°.
So, that means geometrically,
432
00:32:17 --> 00:32:21
my two vectors are going more
or less in the same direction.
433
00:32:21 --> 00:32:27
They make an acute angle.
It's going to be zero if the
434
00:32:27 --> 00:32:33
angle is exactly 90°,
OK, because that's when the
435
00:32:33 --> 00:32:39
cosine will be zero.
And, it will be negative if the
436
00:32:39 --> 00:32:43
angle is more than 90°.
So, that means they go,
437
00:32:43 --> 00:32:46
however, in opposite
directions.
438
00:32:46 --> 00:32:50
So, that's basically one way to
think about what dot product
439
00:32:50 --> 00:32:54
measures.
It measures how much the two
440
00:32:54 --> 00:32:58
vectors are going along each
other.
441
00:32:58 --> 00:33:02
OK, and that actually leads us
to the next application.
442
00:33:02 --> 00:33:05
So, let's see,
did I have a number one there?
443
00:33:05 --> 00:33:07
Yes.
So, if I had a number one,
444
00:33:07 --> 00:33:12
I must have number two.
The second application is to
445
00:33:12 --> 00:33:16
detect orthogonality.
It's to figure out when two
446
00:33:16 --> 00:33:21
things are perpendicular.
OK, so orthogonality is just a
447
00:33:21 --> 00:33:26
complicated word from Greek to
say things are perpendicular.
448
00:33:26 --> 00:33:34
So, let's just take an example.
Let's say I give you the
449
00:33:34 --> 00:33:41
equation x 2y 3z = 0.
OK, so that defines a certain
450
00:33:41 --> 00:33:46
set of points in space,
and what do you think the set
451
00:33:46 --> 00:33:52
of solutions look like if I give
you this equation?
452
00:33:52 --> 00:34:01
So far I see one,
two, three answers,
453
00:34:01 --> 00:34:06
OK.
So, I see various competing
454
00:34:06 --> 00:34:11
answers, but,
yeah, I see a lot of people
455
00:34:11 --> 00:34:18
voting for answer number four.
I see also some I don't knows,
456
00:34:18 --> 00:34:22
and some other things.
But, the majority vote seems to
457
00:34:22 --> 00:34:26
be a plane.
And, indeed that's the correct
458
00:34:26 --> 00:34:28
answer.
So, how do we see that it's a
459
00:34:28 --> 00:34:28
plane?
460
00:34:28 --> 00:34:43
461
00:34:43 --> 00:34:49
So, I should say,
this is the equation of a
462
00:34:49 --> 00:34:52
plane.
So, there's many ways to see
463
00:34:52 --> 00:34:55
that, and I'm not going to give
you all of them.
464
00:34:55 --> 00:34:58
But, here's one way to think
about it.
465
00:34:58 --> 00:35:03
So, let's think geometrically
about how to express this
466
00:35:03 --> 00:35:09
condition in terms of vectors.
So, let's take the origin O,
467
00:35:09 --> 00:35:13
by convention is the point
(0,0,0).
468
00:35:13 --> 00:35:18
And, let's take a point,
P, that will satisfy this
469
00:35:18 --> 00:35:21
equation on it,
so, at coordinates x,
470
00:35:21 --> 00:35:24
y, z.
So, what does this condition
471
00:35:24 --> 00:35:28
here mean?
Well, it means the following
472
00:35:28 --> 00:35:32
thing.
So, let's take the vector, OP.
473
00:35:32 --> 00:35:37
OK, so vector OP,
of course, has components x,
474
00:35:37 --> 00:35:40
y, z.
Now, we can think of this as
475
00:35:40 --> 00:35:44
actually a dot product between
OP and a mysterious vector that
476
00:35:44 --> 00:35:47
won't remain mysterious for very
long,
477
00:35:47 --> 00:35:50
namely, the vector one,
two, three.
478
00:35:50 --> 00:35:59
OK, so, this condition is the
same as OP.A equals zero,
479
00:35:59 --> 00:36:03
right?
If I take the dot product
480
00:36:03 --> 00:36:09
OPdotA I get x times one plus y
times two plus z times three.
481
00:36:09 --> 00:36:14
But now, what does it mean that
the dot product between OP and A
482
00:36:14 --> 00:36:19
is zero?
Well, it means that OP and A
483
00:36:19 --> 00:36:25
are perpendicular.
OK, so I have this vector, A.
484
00:36:25 --> 00:36:28
I'm not going to be able to
draw it realistically.
485
00:36:28 --> 00:36:32
Let's say it goes this way.
Then, a point,
486
00:36:32 --> 00:36:37
P, solves this equation exactly
when the vector from O to P is
487
00:36:37 --> 00:36:40
perpendicular to A.
And, I claim that defines a
488
00:36:40 --> 00:36:41
plane.
For example,
489
00:36:41 --> 00:36:45
if it helps you to see it,
take a vertical vector.
490
00:36:45 --> 00:36:47
What does it mean to be
perpendicular to the vertical
491
00:36:47 --> 00:36:49
vector?
It means you are horizontal.
492
00:36:49 --> 00:36:56
It's the horizontal plane.
Here, it's a plane that passes
493
00:36:56 --> 00:37:05
through the origin and is
perpendicular to this vector,
494
00:37:05 --> 00:37:14
A.
OK, so what we get is a plane
495
00:37:14 --> 00:37:25
through the origin perpendicular
to A.
496
00:37:25 --> 00:37:29
And, in general,
what you should remember is
497
00:37:29 --> 00:37:35
that two vectors have a dot
product equal to zero if and
498
00:37:35 --> 00:37:41
only if that's equivalent to the
cosine of the angle between them
499
00:37:41 --> 00:37:46
is zero.
That means the angle is 90°.
500
00:37:46 --> 00:37:51
That means A and B are
perpendicular.
501
00:37:51 --> 00:37:57
So, we have a very fast way of
checking whether two vectors are
502
00:37:57 --> 00:38:01
perpendicular.
So, one additional application
503
00:38:01 --> 00:38:05
I think we'll see actually
tomorrow is to find the
504
00:38:05 --> 00:38:10
components of a vector along a
certain direction.
505
00:38:10 --> 00:38:13
So, I claim we can use this
intuition I gave about dot
506
00:38:13 --> 00:38:16
product telling us how much to
vectors go in the same direction
507
00:38:16 --> 00:38:19
to actually give a precise
meaning to the notion of
508
00:38:19 --> 00:38:22
component for vector,
not just along the x,
509
00:38:22 --> 00:38:27
y, or z axis,
but along any direction in
510
00:38:27 --> 00:38:31
space.
So, I think I should probably
511
00:38:31 --> 00:38:34
stop here.
But, I will see you tomorrow at
512
00:38:34 --> 00:38:38
2:00 here, and we'll learn more
about that and about cross
513
00:38:38 --> 00:38:44
products.
514
00:38:44 --> 00:38:49