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The bad news today is that there
will be quite a bit of math.

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But the good news
is that we will only do it once

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and it will only take
something like half-hour.

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There are quantities in physics

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which are determined uniquely
by one number.

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Mass is one of them.

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Temperature is one of them.

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Speed is one of them.

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We call those scalars.

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There are others where
you need more than one number

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for instance, on a one-
dimensional motion, velocity

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it has a certain magnitude--
that's the speed--

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but you also have to know

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whether it goes this way
or that way.

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So there has to be a direction.

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Velocity is a vector and
acceleration is a vector

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and today we're going to learn
how to work with these vectors.

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A vector has a length
and a vector has a direction

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and that's why we actually
represent it by an arrow.

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We all have seen...
this is a vector.

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Remember this--
this is a vector.

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If you look at the vector
head-on, you see a dot.

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If you look at the vector
from behind, you see a cross.
 

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This is a vector

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and that will be
our representation of vectors.

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Imagine that I am standing
on the table in 26.100.

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This is the table and I am
standing, say, at point O

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and I move along a straight line
from O to point P

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so I move like so.

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That's why I am on the table

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and that's where you will see me
when you look from 26.100.

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It just so happens

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that someone is also going
to move the table--

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in that same amount of time--
from here to there.

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So that means that the table
will have moved down

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and so my point P will have
moved down exactly the same way

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and so you will see me
now at point S.

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You will see me
at point S in 26.100

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although I am still standing

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at the same location
on the table.

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The table has moved.

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This is now the position
of the table.

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See, the whole table
has shifted.

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Now, if these two motions
take place simultaneously

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then what you will see
from where you are sitting...

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you will see me move in 26.100
from O straight line to S

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and this holds the secret
behind the adding of vectors.

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We say here that the vector OS--
we'll put an arrow over it--

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is the vector OP, with an arrow
over it, plus PS.

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This defines how we add vectors.

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There are various ways
that you can add vectors.

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Suppose I have here vector A
and I have here vector B.

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Then you can do it this way

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which I call
the "head-tail" technique.

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I take B and I bring it
to the head of A.

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So this is B, this is a vector

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and then the net result
is A plus B.

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This vector C equals A plus B.

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That's one way of doing it.

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It doesn't matter
whether you take B...

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the tail of B to the head of A

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or whether you take the tail of
A and bring it to the head of B.

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You will get the same result.

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There's another way
you can do it

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and I call that
"the parallelogram method."

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Here you have A.

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You bring the two tails
together, so here is B now

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so the tails are touching

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and now you complete
this parallelogram.

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And now this vector C
is the same sum vector

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that you have here,
whichever way you prefer.

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You see immediately

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that A plus B is the same
as B plus A.

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There is no difference.

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What is the meaning
of a negative vector?

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Well, A minus A equals zero--

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vector A subtract
from vector A equals zero.

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So here is vector A.

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So which vector do I have
to add to get zero?

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I have to add minus A.

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Well, if you use
the head-tail technique...

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This is A.

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You have to add this vector
to have zero

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so this is minus A

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and so minus A is nothing
but the same as A

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but flipped over 180 degrees.

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We'll use that very often.

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And that brings us to the point
of subtraction of vectors.

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How do we subtract vectors?

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So A minus B equals C.

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Here we have vector A
and here we have--

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let me write this down here--
and here we have vector B.

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One way to look at this
is the following.

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You can say A minus B
is A plus minus B

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and we know how to add vectors
and we know what minus B is.

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Minus B is the same vector
but flipped over

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so we put here minus B

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and so this vector now
here equals A minus B.

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Here's vector C,
here's A minus B.

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And, of course, you can do it
in different ways.

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You can also think of it
as A plus... as C plus B is A.

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Right? You can say you can
bring this to the other side.

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You can say C plus B is A,
C plus B is A.

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In other words, which vector
do I have to add to B to get A?

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And then you have the
parallelogram technique again.

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There are many ways
you can do it.

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The head-tail technique

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is perhaps the easiest
and the safest.

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So you can add
a countless number of vectors

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one plus the other,
and the next one

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and you finally have the sum
of five or six or seven vectors

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which, then, can be
represented by only one.

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When you add scalars,
for instance, five and four

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then there is only one answer,
that is nine.

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Five plus four is nine.

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Suppose you have two vectors.

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You have no information
on their direction

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but you do know that the
magnitude of one is four

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and the magnitude
of the other is five.

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That's all you know.

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Then the magnitude of
the sum vector could be nine

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if they are both in the same
direction-- that's the maximum--

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or it could be one, if they
are in opposite directions.

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So then you have
a whole range of possibilities

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because you do not know
the direction.

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So the adding and the
subtraction of vectors

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is way more complicated
than just scalars.

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As we have seen,
that the sum of vectors

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can be represented by one vector

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equally can we take one vector

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and we can replace it
by the sum of others.

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And we call that
"decomposition" of a vector.

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And that's going to be
very important in 801

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and I want you to follow this,
therefore, quite closely.

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I have a vector which is
in three-dimensional space.

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This is my z axis...

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this is my x axis,
y axis and z axis.

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This is the origin O
and here is a point P

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and I have a vector OP--
that's the vector.

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And what I do now,
I project this vector

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onto the three axes,
x, y and z.

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So there we go.

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Each one has her or his
own method of doing this.

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There we are.

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I call this vector
vector A.

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Now, this angle will be theta,
and this angle will be phi.

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Notice that the projection of A
on the y axis has here

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a number which I call A of y.

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This number is A of x and
this number here is A of z--

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simply a projection of
that vector onto the three axes.

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We now introduce
what we call "unit vectors."

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Unit vectors are always pointing
in the direction

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of the positive axis

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and the unit vector in
the x direction is this one.

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It has a length one,
and we write for it "x roof."

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"Roof" always means unit vector.

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And this is the unit vector
in the y direction

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and this is the unit vector
in the z direction.

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And now I'm going
to rewrite vector A

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in terms of the three components
that we have here.

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So the vector A,
I'm going to write

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as "A of x times x roof,
plus A of y times y roof

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plus A of z times z roof."

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And this A of x times x
is really a vector

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that runs from the origin
to this point.

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So we could put in that
as a vector, if you want to.

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This makes it a vector.

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This is that vector.

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A of y times... oh, sorry,
it is A of x, this one.

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A of y times y roof is this one

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and A of z times z roof
is this one.

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And so these three green vectors
added together

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are exactly identical
to the vector OP

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so we have decomposed one vector
into three directions.

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And we will see that very often,
this is of great use in 801.

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The magnitude of the vector is

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the square root of Ax squared
plus Ay squared plus Az squared

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and so we can take
a simple example.

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For instance,
I take a vector A--

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this is just an example,
to see this in action--

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and we call A three X roof,

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so A of axis is three
minus five y roof plus 6 Z roof

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so that means that it's
three units in this direction

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it is five units
in this direction--

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in the minus y direction--

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and six in the plus z direction.

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That makes up a vector
and I call that vector A.

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What is the magnitude
of that vector--

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which I always write down
with vertical bars--

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if I put two bars on one side,
that's always the magnitude

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or sometimes
I simply leave the arrow off,

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but to be always on
the safe side, I like this idea

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that you know it's
really the magnitude

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becomes the scalar
when you do that.

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So that would be the square root
of three squared is nine

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five squared is 25,
six squared is 36

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so that's the square root of 70.

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And suppose I asked you,
"What is theta?"

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It's uniquely determined,
of course.

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This vector
is uniquely determined

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in three-dimensional space

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so you should be able to find
phi and theta.

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Well, the cosine of theta...

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See, this angle here...
90 degrees projection.

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So the cosine of theta
is A of z divided by A itself.

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So the cosine of theta equals
A of z divided by A itself

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which in our case

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would be six divided
by the square root of 70.

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And you can do fine.

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It's just simply a matter
of manipulating some numbers.

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We now come to a much more
difficult part of vectors

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and that is multiplication
of vectors.

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We're not going to need this
until October, but I decided

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we might as well
get it over with now.

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Now that we introduced vectors,
you can add and subtract

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you might as well learn
about multiplication.

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It's sort of, the job is done,
it's like going to the dentist.

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It's a little painful,
but it's good for you

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and when it's behind you,
the pain disappears.

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So we're going to talk about
multiplication of vectors

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something that will not
come back until October

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and later in the course.

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There are two ways
that we multiply vectors

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and one is called
the "dot product"

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often also called
the scalar product.

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A dot B, a fat dot, and that
is defined as it is a scalar.

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A of x times B of x,
just a number

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plus A of y times B of y--
that's another number--

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plus A of z times B of z--
that's another number.

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It is a scalar.

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It has no longer a direction.

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That is the dot product.

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So that's method number one.

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That's completely legitimate
and you can always use that.

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There is another way
to find the dot product

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depending upon
what you're being given--

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how the problem is
presented to you.

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If someone gives you the vector
A and you have the vector B

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and you happen to know
this angle between them,

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this angle theta--

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which has nothing to do
with that angle theta;

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it's the angle between the two--

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then the dot product
is also the following

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and you may make an attempt
to prove that.

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You project the vector B on A.

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This is that projection.

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The length of this vector
is B cosine theta.

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And then the dot product

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is the magnitude of A
times the magnitude of B

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times the cosine
of the angle theta.

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The two are
completely identical.

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Now, you may ask me,
you may say,

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"Gee, how do I know
what theta is?

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"How do I know
I should take theta this angle

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"or maybe I should take
theta this angle?

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I mean, what angle
is A making with B?"

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It makes no difference

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because the cosine
of this angle here

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is the same as the cosine
of 360 degrees minus theta

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so that makes no difference.

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Sometimes this is faster

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depending upon how the problem
is presented to you;

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sometimes the other is faster.

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You can immediately see
by looking at this--

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it's easier to see
than looking here--

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that the dot product can be
larger than zero

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it can be equal to zero
and it can be smaller than zero.

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A and B are, by definition,
always positive.

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They are a magnitude.

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That's always determined
by the cosine of theta.

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If the cosine of theta
is larger than zero,

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well, then
it's larger than zero.

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The cosine of theta can be zero.

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If the angle for theta
is pi over two--

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in other words,
if the two vectors

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are perpendicular
to each other--

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then the dot product is zero,
and if this angle theta

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00:17:20 --> 00:17:26
is between 90 degrees
and 180 degrees

284
00:17:23 --> 00:17:29
then the cosine is negative.

285
00:17:26 --> 00:17:32
We will see that at work,
no pun implied

286
00:17:29 --> 00:17:35
when we're going to deal
with work in physics.

287
00:17:32 --> 00:17:38
You will see
that we can do positive work

288
00:17:34 --> 00:17:40
and we can do negative work

289
00:17:36 --> 00:17:42
and that has to do
with this dot product.

290
00:17:38 --> 00:17:44
Work and energy are
dot products.

291
00:17:43 --> 00:17:49
I could do an extremely
simple example with you;

292
00:17:48 --> 00:17:54
the simplest
that I can think of.

293
00:17:51 --> 00:17:57
Perhaps it's almost an insult--
it's not meant that way.

294
00:17:56 --> 00:18:02
Suppose we have A dot B

295
00:18:01 --> 00:18:07
and A is the one that you really
have on the blackboard there.

296
00:18:06 --> 00:18:12
Right here, that's A.

297
00:18:08 --> 00:18:14
But B is just two y roof.

298
00:18:14 --> 00:18:20
Two y roof, that's all it is.

299
00:18:19 --> 00:18:25
Well, what is A dot B?

300
00:18:21 --> 00:18:27
A dot B... there's
no x component of B

301
00:18:26 --> 00:18:32
so that becomes zero,
this term becomes zero.

302
00:18:29 --> 00:18:35
There is only
a y component of B

303
00:18:31 --> 00:18:37
so it is minus five
times plus two

304
00:18:35 --> 00:18:41
so I get minus ten, because
there was no z component.

305
00:18:39 --> 00:18:45
Simple as that,
so it's minus ten.

306
00:18:43 --> 00:18:49
I can give you another example,
example two.

307
00:18:48 --> 00:18:54
Suppose A itself is the unit
vector in the y direction

308
00:18:53 --> 00:18:59
and B is the unit vector
in the z direction.

309
00:18:59 --> 00:19:05
Then A dot B is what?

310
00:19:05 --> 00:19:11
I want to hear it
loud and clear.

311
00:19:07 --> 00:19:13
CLASS:
Zero.

312
00:19:08 --> 00:19:14
LEWIN:
Yeah! Zero.

313
00:19:10 --> 00:19:16
It is zero-- you don't even
have to think about anything.

314
00:19:13 --> 00:19:19
You know that these two
are at 90 degrees.

315
00:19:15 --> 00:19:21
If you want to waste your time

316
00:19:17 --> 00:19:23
and want to substitute it
in here

317
00:19:19 --> 00:19:25
you will see
that it comes out to be zero.

318
00:19:21 --> 00:19:27
It should work, because clearly
A of y means

319
00:19:24 --> 00:19:30
that this... this is one.

320
00:19:28 --> 00:19:34
That's what it means.

321
00:19:29 --> 00:19:35
And B is z, that means
that B of z... this is one

322
00:19:33 --> 00:19:39
and all the others do not exist.

323
00:19:36 --> 00:19:42
Well, I wish you luck
with that and we now go

324
00:19:39 --> 00:19:45
to a way more difficult
part of multiplication

325
00:19:43 --> 00:19:49
and that is
vector multiplication

326
00:19:46 --> 00:19:52
which is called
"the vector product."

327
00:19:52 --> 00:19:58
Or also called...
most of the time

328
00:19:54 --> 00:20:00
I refer to it
as "the cross product."

329
00:19:58 --> 00:20:04
The cross product is written
like so: A cross B equals C.

330
00:20:06 --> 00:20:12
It's a cross, very clear cross.

331
00:20:10 --> 00:20:16
And I will tell you
how I remember...

332
00:20:12 --> 00:20:18
that is, method number one.

333
00:20:13 --> 00:20:19
I'm going to teach you--
just like with the dot product--

334
00:20:15 --> 00:20:21
two methods.

335
00:20:16 --> 00:20:22
I will tell you
method number one

336
00:20:18 --> 00:20:24
which is the one
that always works.

337
00:20:20 --> 00:20:26
It's time-consuming,
but it always works.

338
00:20:23 --> 00:20:29
You write down here
a matrix with three rows.

339
00:20:27 --> 00:20:33
The first row is x roof,
y roof, z roof.

340
00:20:33 --> 00:20:39
The second one is
A of x, A of y, A of z.

341
00:20:38 --> 00:20:44
It's important,
if A is here first

342
00:20:40 --> 00:20:46
that that second row must be A
and the third row is then B.

343
00:20:45 --> 00:20:51
B of x, B of y, B of z.

344
00:20:49 --> 00:20:55
So these six are numbers
and these are the unit vectors.

345
00:20:54 --> 00:21:00
I repeat this here verbatim--

346
00:21:00 --> 00:21:06
you will see in a minute
why I need that--

347
00:21:07 --> 00:21:13
and I will do the same here.

348
00:21:09 --> 00:21:15

349
00:21:17 --> 00:21:23
Okay, and now comes the recipe.

350
00:21:19 --> 00:21:25
You take... you go from
the upper left-hand corner

351
00:21:24 --> 00:21:30
to the one in this direction.

352
00:21:28 --> 00:21:34
You multiply them, all three,
and that's a plus sign.

353
00:21:32 --> 00:21:38
So you get Ay... so C

354
00:21:35 --> 00:21:41
which is A cross B equals Ay,
times Bz, times the x roof--

355
00:21:45 --> 00:21:51
but I'm not going to put
the x roof in yet--

356
00:21:48 --> 00:21:54
because I have to subtract
this one... minus sign

357
00:21:55 --> 00:22:01
which has Az By

358
00:21:58 --> 00:22:04
so it is minus Az By, and
that is in the direction x.

359
00:22:07 --> 00:22:13
The next one is this one.

360
00:22:11 --> 00:22:17
Az Bx...

361
00:22:17 --> 00:22:23
minus this one

362
00:22:23 --> 00:22:29
Ax Bz

363
00:22:29 --> 00:22:35
in the direction y.

364
00:22:32 --> 00:22:38
And last but not least

365
00:22:35 --> 00:22:41
Ax By...

366
00:22:42 --> 00:22:48
minus Ay Bx...

367
00:22:52 --> 00:22:58
in the direction
of the unit vector z.

368
00:22:59 --> 00:23:05
So this part here is
what we call "C of x".

369
00:23:04 --> 00:23:10
It's the x component
of this vector

370
00:23:07 --> 00:23:13
and this we can call "C of y"
and this we can call "C of z."

371
00:23:13 --> 00:23:19
So we can also write
that vector, then,

372
00:23:16 --> 00:23:22
that C equals C of x, x roof,
plus C of y, y roof

373
00:23:23 --> 00:23:29
plus C of z, z roof.

374
00:23:27 --> 00:23:33
Cross product of A and B.

375
00:23:31 --> 00:23:37
We will have lots of exercises,

376
00:23:33 --> 00:23:39
lots of chances you will have
on assignment, too

377
00:23:36 --> 00:23:42
to play with this a little bit.

378
00:23:37 --> 00:23:43
Now comes my method number two
and method number two is, again

379
00:23:42 --> 00:23:48
as we had with the dot product,
is a geometrical method.

380
00:23:51 --> 00:23:57
Let me try to work
on this board in between.

381
00:23:56 --> 00:24:02
If you know vector A
and you know vector B

382
00:24:05 --> 00:24:11
and you know
that the angle is theta

383
00:24:08 --> 00:24:14
then the cross product, C,
equals A cross B

384
00:24:14 --> 00:24:20
is the magnitude of A
times the magnitude of B

385
00:24:19 --> 00:24:25
times the sine of theta

386
00:24:21 --> 00:24:27
not the cosine of theta as we
had before with the dot product.

387
00:24:25 --> 00:24:31
It is the sine of theta.

388
00:24:29 --> 00:24:35
So you can really immediately
see that this will be zero

389
00:24:32 --> 00:24:38
if theta is either zero degrees
or 180 degrees

390
00:24:36 --> 00:24:42
whereas the dot product was zero

391
00:24:37 --> 00:24:43
when the angle between them
was 90 degrees.

392
00:24:43 --> 00:24:49
This number can be larger
than zero

393
00:24:45 --> 00:24:51
if the sine theta
is larger than zero.

394
00:24:46 --> 00:24:52
It can also be smaller
than zero.

395
00:24:48 --> 00:24:54
Now we only have
the magnitude of the vector

396
00:24:51 --> 00:24:57
and now comes the hardest part.

397
00:24:53 --> 00:24:59
What is the direction
of the vector?

398
00:24:55 --> 00:25:01
And that is something

399
00:24:57 --> 00:25:03
that you have to engrave
in your mind and not forget.

400
00:25:01 --> 00:25:07
The direction is found
as follows.

401
00:25:04 --> 00:25:10
You take A,
because it's first mentioned

402
00:25:08 --> 00:25:14
and you rotate A over the
shortest possible angle to B.

403
00:25:13 --> 00:25:19
If you had in your hand
a corkscrew--

404
00:25:15 --> 00:25:21
and I will show that
in a minute--

405
00:25:16 --> 00:25:22
then you turn the corkscrew as
seen from your seats clockwise

406
00:25:20 --> 00:25:26
and the corkscrew would go
into the blackboard.

407
00:25:23 --> 00:25:29
And if the corkscrew goes
into the blackboard

408
00:25:25 --> 00:25:31
you will see the tail
of the vector

409
00:25:27 --> 00:25:33
and you will see a cross,
little plus sign

410
00:25:30 --> 00:25:36
and therefore we put that
like so.

411
00:25:34 --> 00:25:40
A cross product is always
perpendicular to both A and B

412
00:25:40 --> 00:25:46
but it leaves you
with two choices:

413
00:25:41 --> 00:25:47
It can either come
out of the blackboard

414
00:25:43 --> 00:25:49
or it can go
in the blackboard

415
00:25:46 --> 00:25:52
and I just told you
which convention to use.

416
00:25:50 --> 00:25:56
And I want to show that to you

417
00:25:52 --> 00:25:58
in a way that may appeal
to you more.

418
00:25:56 --> 00:26:02
This is what I have used before

419
00:25:59 --> 00:26:05
on my television help sessions

420
00:26:05 --> 00:26:11
that I have given at MIT.

421
00:26:07 --> 00:26:13
I have an apple--
not an apple...

422
00:26:08 --> 00:26:14
This is a tomato--
not a tomato...

423
00:26:10 --> 00:26:16
It's a potato.

424
00:26:11 --> 00:26:17
(class laughs )

425
00:26:12 --> 00:26:18
I have a potato here
and here is a corkscrew.

426
00:26:15 --> 00:26:21
Here is a corkscrew.

427
00:26:17 --> 00:26:23
I'm going to turn the corkscrew

428
00:26:19 --> 00:26:25
as seen from your side,
clockwise.

429
00:26:22 --> 00:26:28
And you'll see that the
corkscrew goes into the potato

430
00:26:29 --> 00:26:35
in -- that's the direction,
then, of the vector.

431
00:26:33 --> 00:26:39
If we had B cross A,
then you take B in your hands

432
00:26:38 --> 00:26:44
and you rotate it over
the shortest angle to A.

433
00:26:40 --> 00:26:46
Now you have to rotate
counterclockwise

434
00:26:43 --> 00:26:49
and when you rotate
counterclockwise

435
00:26:45 --> 00:26:51
the corkscrew comes to you--
there you go--

436
00:26:48 --> 00:26:54
and so the vector is now
pointing in this direction.

437
00:26:51 --> 00:26:57
And if the vector
is pointing towards you

438
00:26:54 --> 00:27:00
then we would indicate that
with a circle and a dot.

439
00:26:58 --> 00:27:04
In other words, for this vector

440
00:27:00 --> 00:27:06
B cross A would have
exactly the same magnitude--

441
00:27:04 --> 00:27:10
no difference-- but it would
be coming out of the blackboard.

442
00:27:09 --> 00:27:15
In other words, A cross B
equals minus B cross A

443
00:27:21 --> 00:27:27
whereas A dot B
is the same as B dot A.

444
00:27:25 --> 00:27:31
We will encounter cross products
when we deal with torques

445
00:27:29 --> 00:27:35
and when we deal
with angular momentum

446
00:27:31 --> 00:27:37
which is not
the easiest part of 801.

447
00:27:35 --> 00:27:41
Let's take
an extremely simple example.

448
00:27:39 --> 00:27:45
Again, I don't mean to insult
you with such a simple example

449
00:27:44 --> 00:27:50
but you will get chances,

450
00:27:46 --> 00:27:52
more advanced chances
on your assignment.

451
00:27:49 --> 00:27:55
Suppose I gave
to vector A this x roof.

452
00:27:54 --> 00:28:00
It's a unit vector
in the x direction.

453
00:27:57 --> 00:28:03
That means A of x is one

454
00:28:00 --> 00:28:06
and A of y is zero
and A of z is zero.

455
00:28:04 --> 00:28:10
And suppose B is y roof.

456
00:28:09 --> 00:28:15
That means B of y is one

457
00:28:13 --> 00:28:19
and B of x is zero
and B of z is zero.

458
00:28:17 --> 00:28:23
What, now, is the dot product,
the cross product, A cross B?

459
00:28:23 --> 00:28:29

460
00:28:26 --> 00:28:32
Well, you can apply that recipe

461
00:28:31 --> 00:28:37
but it's much easier to go

462
00:28:34 --> 00:28:40
to the x, y, z axes
that we have here.

463
00:28:39 --> 00:28:45
A was in the x direction,
the unit vector

464
00:28:41 --> 00:28:47
and B in the y direction.

465
00:28:44 --> 00:28:50
I take A in my hand, I rotate
over the smallest angle

466
00:28:47 --> 00:28:53
which is 90 degrees to y,
and my corkscrew will go up.
 

467
00:28:51 --> 00:28:57
So I know
the whole thing already.

468
00:28:53 --> 00:28:59
I know that this cross product
must be z roof.

469
00:28:58 --> 00:29:04
The magnitude must be one.

470
00:29:00 --> 00:29:06
That's immediately clear.

471
00:29:01 --> 00:29:07
But I immediately have
the direction

472
00:29:04 --> 00:29:10
by using the corkscrew rule.

473
00:29:06 --> 00:29:12
Now if you're very smart

474
00:29:09 --> 00:29:15
you may say,
"Aha! You find plus z

475
00:29:14 --> 00:29:20
"only because you have used
this coordinate system.

476
00:29:16 --> 00:29:22
"If this axis had been x,
and this one had been y

477
00:29:22 --> 00:29:28
"then the cross product
of x and y would be

478
00:29:24 --> 00:29:30
in the minus z direction."

479
00:29:26 --> 00:29:32
Yeah, you're right.

480
00:29:28 --> 00:29:34
But if you ever do that,
I willkill you!

481
00:29:31 --> 00:29:37
(class laughs )

482
00:29:32 --> 00:29:38
You will always,
always have to work

483
00:29:34 --> 00:29:40
with what we call "a right-
handed coordinate system."

484
00:29:38 --> 00:29:44
And a right-handed coordinate
system, by definition

485
00:29:41 --> 00:29:47
is one whereby the cross product

486
00:29:44 --> 00:29:50
of x with y is z
and not y minus z.

487
00:29:48 --> 00:29:54
So whenever you get,
in the future, involved

488
00:29:50 --> 00:29:56
with cross products and torques
and angular momentum

489
00:29:52 --> 00:29:58
always make yourself
an xyz diagram

490
00:29:56 --> 00:30:02
for which x cross y is z.

491
00:29:59 --> 00:30:05
Never, ever make it such
that x cross y is minus z.

492
00:30:02 --> 00:30:08
You're going to hang yourself.

493
00:30:04 --> 00:30:10
For one thing,
that wouldn't work anymore.

494
00:30:08 --> 00:30:14
So be very, very careful.

495
00:30:10 --> 00:30:16
You must work... if you use
the right-hand corkscrew rule

496
00:30:12 --> 00:30:18
make sure you work with the
right-handed coordinate system.

497
00:30:19 --> 00:30:25
All right,
now the worst part is over.

498
00:30:24 --> 00:30:30
And now I would like
to write down for you...

499
00:30:30 --> 00:30:36
We pick up some
of the fruits now

500
00:30:33 --> 00:30:39
although it will
penetrate slowly.

501
00:30:36 --> 00:30:42
I want to write down for you
equations for a moving particle

502
00:30:42 --> 00:30:48
a moving object
in three-dimensional space--

503
00:30:48 --> 00:30:54
very complicated motion

504
00:30:49 --> 00:30:55
which I can hardly imagine
what it's like.

505
00:30:55 --> 00:31:01
It is a point that is going
to move around in space

506
00:30:59 --> 00:31:05
and it is this point P

507
00:31:02 --> 00:31:08
this point P is going
to move around in space

508
00:31:05 --> 00:31:11
and I call this vector OP,
I call that now vector r

509
00:31:11 --> 00:31:17
and I give it a sub-index t

510
00:31:13 --> 00:31:19
which indicates
it's changing with time.

511
00:31:16 --> 00:31:22
I call this location A of y,
I am going to call that y of t.

512
00:31:21 --> 00:31:27
It's changing with time.

513
00:31:23 --> 00:31:29
I call this x of t--
it's going to change with time--

514
00:31:26 --> 00:31:32
and I call this point z of t

515
00:31:29 --> 00:31:35
which is going
to change with time

516
00:31:31 --> 00:31:37
because point P
is going to move.

517
00:31:34 --> 00:31:40
And so I'm going to write down
the vector r

518
00:31:38 --> 00:31:44
in its most general form
that I can do that.

519
00:31:41 --> 00:31:47
R, which changes with time

520
00:31:43 --> 00:31:49
is now x of t-- which is the
same as a over x there, before--

521
00:31:49 --> 00:31:55
times x roof plus y of t,

522
00:31:55 --> 00:32:01
y roof plus z of t, z roof.

523
00:32:00 --> 00:32:06
I have decomposed my vector r
into three independent vectors.

524
00:32:06 --> 00:32:12
Each one of those change
with time.

525
00:32:10 --> 00:32:16
What is the velocity
of this particle?

526
00:32:12 --> 00:32:18
Well, the velocity is the first
derivative of the position

527
00:32:18 --> 00:32:24
so that it is dr dt.

528
00:32:22 --> 00:32:28
So there we go-- first
the derivative of this one

529
00:32:26 --> 00:32:32
which is dx dt, x roof.

530
00:32:30 --> 00:32:36
I am going to write for dx dt
"x dot," because I am lazy

531
00:32:36 --> 00:32:42
and I am going to write for
d2x dt squared, "x double dots."

532
00:32:41 --> 00:32:47
It's often done,
but not in your book.

533
00:32:43 --> 00:32:49
But it is a notation
that I will often use

534
00:32:45 --> 00:32:51
because otherwise the equations
look so clumsy.

535
00:32:48 --> 00:32:54
Plus y dot times y roof
plus z dot times z roof.

536
00:32:56 --> 00:33:02
So z dot is the dz/dt.

537
00:32:59 --> 00:33:05
What is the acceleration
as a function of time?

538
00:33:02 --> 00:33:08
Well, the acceleration
as a function of time

539
00:33:05 --> 00:33:11
equals dv/dt.

540
00:33:08 --> 00:33:14
So that's the section derivative
of x versus time

541
00:33:12 --> 00:33:18
and so that becomes
x double dot times x roof

542
00:33:17 --> 00:33:23
plus y double dot times y roof
plus z double dot times z roof.

543
00:33:26 --> 00:33:32
And look
what we have now accomplished.

544
00:33:30 --> 00:33:36
It looks like minor, but
it's going to be big later on.

545
00:33:33 --> 00:33:39
We have a point P going
in three-dimensional space

546
00:33:38 --> 00:33:44
and here we have the entire
behavior of the object

547
00:33:44 --> 00:33:50
as it moves its projection
along the x axis.

548
00:33:49 --> 00:33:55
This is the position,
this is its velocity

549
00:33:53 --> 00:33:59
and this is its acceleration.

550
00:33:55 --> 00:34:01
And here you can see the
entire behavior on the z axis.

551
00:34:01 --> 00:34:07
This is the position
on the z axis

552
00:34:03 --> 00:34:09
this is the velocity component
in the z direction

553
00:34:05 --> 00:34:11
and this is the acceleration
on the z axis.

554
00:34:08 --> 00:34:14
And here you have the y.

555
00:34:09 --> 00:34:15
In other words, we have now...
the three-dimensional motion

556
00:34:13 --> 00:34:19
we have cut into three
one-dimensional motions.

557
00:34:19 --> 00:34:25
This is
a one-dimensional motion.

558
00:34:21 --> 00:34:27
This is behavior
only along the x axis

559
00:34:24 --> 00:34:30
and this is a behavior
only along the y axis

560
00:34:26 --> 00:34:32
and this is a behavior
only along the z axis

561
00:34:29 --> 00:34:35
and the three together make up

562
00:34:33 --> 00:34:39
the actual motion
of that particle.

563
00:34:37 --> 00:34:43
What have we gained now?

564
00:34:38 --> 00:34:44
It looks like... this looks
like a mathematical zoo.

565
00:34:41 --> 00:34:47
You would say, "Well, if this
is what it is going to be like

566
00:34:44 --> 00:34:50
it's going to be hell."

567
00:34:45 --> 00:34:51
Well, not quite--

568
00:34:49 --> 00:34:55
in fact, it's going
to help you a great deal.

569
00:34:53 --> 00:34:59
First of all, if I throw up
a tennis ball in class

570
00:34:56 --> 00:35:02
like this, then
the whole trajectory is...

571
00:35:02 --> 00:35:08
the whole trajectory is
in one plane

572
00:35:04 --> 00:35:10
in the vertical plane.

573
00:35:06 --> 00:35:12
So even though it is
in three dimensions

574
00:35:08 --> 00:35:14
we can always represent it by
two axes, by two dimensionally

575
00:35:12 --> 00:35:18
a y axis and an x axis

576
00:35:14 --> 00:35:20
so already
the three-dimensional problem

577
00:35:16 --> 00:35:22
often becomes
a two-dimensional problem.

578
00:35:20 --> 00:35:26
We will, with great success,
analyze these trajectories

579
00:35:25 --> 00:35:31
by decomposing
this very complicated motion.

580
00:35:28 --> 00:35:34
Imagine what an incredibly
complicated arc that is

581
00:35:31 --> 00:35:37
and yet we are going
to decompose it

582
00:35:33 --> 00:35:39
into a motion
in the x direction

583
00:35:36 --> 00:35:42
which lives a life of its own

584
00:35:38 --> 00:35:44
independent of the motion
in the y direction

585
00:35:40 --> 00:35:46
which lives a life ofits own

586
00:35:42 --> 00:35:48
and, of course, you always have
to combine the two

587
00:35:44 --> 00:35:50
to know what the particle
is doing.

588
00:35:51 --> 00:35:57
We know the equations so well
from our last lecture

589
00:35:56 --> 00:36:02
from one-dimensional motion
with constant acceleration.

590
00:36:03 --> 00:36:09
The first line tells you

591
00:36:04 --> 00:36:10
what the x position is
as a function of time.

592
00:36:07 --> 00:36:13
The index t tells you
that it is changing with time.

593
00:36:11 --> 00:36:17
It is the position
at t equals zero

594
00:36:13 --> 00:36:19
plus the velocity
at t equals zero

595
00:36:16 --> 00:36:22
times t plus
one-half ax t squared

596
00:36:19 --> 00:36:25
if there is an acceleration
in the x direction.

597
00:36:21 --> 00:36:27
The velocity immediately comes

598
00:36:23 --> 00:36:29
from taking the derivative
of this function

599
00:36:25 --> 00:36:31
and the acceleration comes

600
00:36:26 --> 00:36:32
from taking the derivative
of this function.

601
00:36:29 --> 00:36:35
Now, if we have a motion
which is more complicated--

602
00:36:34 --> 00:36:40
which reaches out to two
or three dimensions--

603
00:36:37 --> 00:36:43
we can decompose the motion
in three perpendicular axes

604
00:36:41 --> 00:36:47
and you can replace
every x here by a y

605
00:36:44 --> 00:36:50
which gives you the entire
behavior in the y direction

606
00:36:48 --> 00:36:54
and if you want to know
the behavior in the z direction

607
00:36:50 --> 00:36:56
you replace every x here by z

608
00:36:53 --> 00:36:59
and then you have decomposed
the motion in three directions.

609
00:36:59 --> 00:37:05
Each of them are linear.

610
00:37:04 --> 00:37:10
And that's
what I want to do now.

611
00:37:06 --> 00:37:12
I'm going to throw up an object,
golf ball or an apple in 26.100

612
00:37:21 --> 00:37:27
and we know that it's in the
vertical plane, so we have...

613
00:37:24 --> 00:37:30
we only deal
with a two-dimensional problem

614
00:37:27 --> 00:37:33
this being...

615
00:37:29 --> 00:37:35
I call this my x axis and I'm
going to call this my y axis.

616
00:37:35 --> 00:37:41
I call this
increasing value of x

617
00:37:38 --> 00:37:44
and I call this
increasing value of y.

618
00:37:42 --> 00:37:48
I could have calledthis
increasing value of y.

619
00:37:45 --> 00:37:51
Today I have decided to call
this increasing value of y.

620
00:37:49 --> 00:37:55
I am free in that choice.

621
00:37:52 --> 00:37:58
I throw up an object
at a certain angle

622
00:37:56 --> 00:38:02
and I see a motion
like this-- boing!--

623
00:37:59 --> 00:38:05
and it comes back to the ground.

624
00:38:04 --> 00:38:10
My initial speed
when I threw it was v zero

625
00:38:12 --> 00:38:18
and the angle here is alpha.

626
00:38:16 --> 00:38:22
The x component
of that initial velocity

627
00:38:22 --> 00:38:28
is v zero cosine alpha

628
00:38:26 --> 00:38:32
and the y component
equals v zero sine alpha.

629
00:38:33 --> 00:38:39
So that's the "begins" velocity
of the x direction.

630
00:38:37 --> 00:38:43
This is the "begins" velocity
in the y direction.

631
00:38:41 --> 00:38:47
A little later in time,
that object is here at point P

632
00:38:51 --> 00:38:57
and this is now
the position vector

633
00:38:54 --> 00:39:00
which we have called r of t,
it's this vector.

634
00:39:02 --> 00:39:08
That's the vector
that is moving through space.

635
00:39:06 --> 00:39:12
At this moment in time,
x of t is here

636
00:39:12 --> 00:39:18
and at this moment in time,
y of t is here.

637
00:39:22 --> 00:39:28
And now you're going to see,
for the first time

638
00:39:26 --> 00:39:32
the big gain by the way that we
have divided the two axes

639
00:39:33 --> 00:39:39
which live an independent life.

640
00:39:35 --> 00:39:41
First x.

641
00:39:37 --> 00:39:43
I want to know everything about
x that there has to be known.

642
00:39:40 --> 00:39:46
I want to know where it is
at any moment in time

643
00:39:44 --> 00:39:50
velocity and the acceleration,
only in x.

644
00:39:48 --> 00:39:54
First I want to know
that at t = 0.

645
00:39:53 --> 00:39:59
Well, at t = 0, I look there

646
00:39:56 --> 00:40:02
X zero-- that's the,
I can choose that to be zero.

647
00:40:00 --> 00:40:06
So I can say x zero is zero,
that's my free choice.

648
00:40:04 --> 00:40:10
Now I need v zero x--
what is the velocity?

649
00:40:08 --> 00:40:14
The velocity at t = 0,
which we have called v zero x

650
00:40:13 --> 00:40:19
is this velocity--
v zero cosine alpha.

651
00:40:17 --> 00:40:23
And it's not going to change.

652
00:40:20 --> 00:40:26
Why is it not going to change?

653
00:40:22 --> 00:40:28
Because there is no a of x,
so this term here is zero

654
00:40:29 --> 00:40:35
we only have this one.

655
00:40:30 --> 00:40:36
So at all moments in time

656
00:40:32 --> 00:40:38
the velocity in the x direction
is v zero cosine alpha

657
00:40:36 --> 00:40:42
and the a of x equals zero.

658
00:40:40 --> 00:40:46

659
00:40:43 --> 00:40:49
Now I want to do the same
in the x direction for time t.

660
00:40:50 --> 00:40:56
Well, at time t, I look there
at the first equation.

661
00:40:55 --> 00:41:01
There it is-- x zero is zero.

662
00:40:58 --> 00:41:04
I know v zero x,
that is v zero cosine alpha

663
00:41:03 --> 00:41:09
so x of t is
v zero cosine alpha times t

664
00:41:09 --> 00:41:15
but there is no acceleration,
so that's it.

665
00:41:13 --> 00:41:19
What is vx of t?

666
00:41:16 --> 00:41:22
The velocity in the x direction
at any moment in time.

667
00:41:20 --> 00:41:26
That is that equation,
that is simply v zero x.

668
00:41:24 --> 00:41:30
It is not changing in time

669
00:41:26 --> 00:41:32
because there is
no acceleration.

670
00:41:29 --> 00:41:35
So the initial velocity
at t zero is the same

671
00:41:32 --> 00:41:38
as t seconds later
and the acceleration is zero.

672
00:41:36 --> 00:41:42
Now we're going to do this
for the y direction.

673
00:41:42 --> 00:41:48
And now you begin to see
the gain for the decomposition.

674
00:41:46 --> 00:41:52
In the y direction,
we change the x by y

675
00:41:51 --> 00:41:57
and so we do it first
at t = 0.

676
00:41:55 --> 00:42:01
So look there.

677
00:41:57 --> 00:42:03
This becomes y zero--
I call that zero.

678
00:42:00 --> 00:42:06
I can always call
my origin zero.

679
00:42:03 --> 00:42:09
I get v zero y times t.

680
00:42:06 --> 00:42:12
Well, v zero y is this quantity

681
00:42:09 --> 00:42:15
is v zero sine alpha,
v zero sine alpha.

682
00:42:19 --> 00:42:25
This is v zero sine alpha.

683
00:42:20 --> 00:42:26
That is the velocity at
time zero, and this is zero.

684
00:42:25 --> 00:42:31
At time zero...
this is zero at time zero.

685
00:42:30 --> 00:42:36
What is the acceleration
in the y direction at time zero?

686
00:42:37 --> 00:42:43
What is the acceleration?
That has to do with gravity.

687
00:42:40 --> 00:42:46
There is no acceleration
in the x direction

688
00:42:43 --> 00:42:49
but you better believe that
there is one in the y direction.

689
00:42:45 --> 00:42:51
So only when we deal
with the y equations

690
00:42:48 --> 00:42:54
does this acceleration come in--

691
00:42:50 --> 00:42:56
not at all when we deal
with the x direction.

692
00:42:53 --> 00:42:59
Well, if we call the
acceleration due to gravity

693
00:42:55 --> 00:43:01
g equals plus 9.80,
and I always call it g

694
00:43:02 --> 00:43:08
what would be the acceleration
in the y direction

695
00:43:04 --> 00:43:10
given the fact that I call this
increasing value of y?

696
00:43:08 --> 00:43:14
CLASS:
Minus 9.8.

697
00:43:12 --> 00:43:18
LEWIN:
Minus 9.8,
which I will also say

698
00:43:15 --> 00:43:21
always call minus g
because my g is always positive.

699
00:43:18 --> 00:43:24
So it is minus g.

700
00:43:22 --> 00:43:28
So that tells the story of t
equals zero in the y direction

701
00:43:26 --> 00:43:32
and now we have to complete it
at time t equals t.

702
00:43:31 --> 00:43:37
At time t equals t,
we have the first line there.

703
00:43:36 --> 00:43:42
Y zero is zero.

704
00:43:38 --> 00:43:44
So we have y as a function
of time, y zero is zero

705
00:43:42 --> 00:43:48
so we don't have to work
with that.

706
00:43:45 --> 00:43:51
Where is my... so this is zero,
so I get v zero y times t

707
00:43:50 --> 00:43:56
so I get
v zero sine alpha times t

708
00:43:58 --> 00:44:04
plus one-half, but it
is minus one-half g t squared

709
00:44:05 --> 00:44:11
and now I get the velocity
in the y direction at time t--

710
00:44:09 --> 00:44:15
that is my second line.

711
00:44:11 --> 00:44:17
That is going to be
v zero sine alpha minus g t

712
00:44:19 --> 00:44:25
and the acceleration in the y
direction at any moment in time

713
00:44:23 --> 00:44:29
equals minus g.

714
00:44:25 --> 00:44:31
And now I have done all I can

715
00:44:27 --> 00:44:33
to completely decompose
this complicated motion

716
00:44:31 --> 00:44:37
into two entirely independent
one-dimensional motions.

717
00:44:36 --> 00:44:42
And the next lecture

718
00:44:38 --> 00:44:44
we're going to use this again
and again and again and again.

719
00:44:41 --> 00:44:47
This lecture is not over yet
but I want you to know

720
00:44:44 --> 00:44:50
that this is
what we're going to apply

721
00:44:46 --> 00:44:52
for many lectures to come--

722
00:44:47 --> 00:44:53
the decomposition
of a complicated trajectory

723
00:44:51 --> 00:44:57
into two simple ones.

724
00:44:55 --> 00:45:01
Now, when you look at this

725
00:44:57 --> 00:45:03
there is something
quite remarkable

726
00:44:59 --> 00:45:05
and the remarkable thing

727
00:45:01 --> 00:45:07
is that the velocity
in the x direction

728
00:45:03 --> 00:45:09
throughout
this whole trajectory--

729
00:45:05 --> 00:45:11
if there is no air draft,
if there is no friction--

730
00:45:07 --> 00:45:13
is not changing.

731
00:45:09 --> 00:45:15
It's only the velocity in the
y direction that is changing.

732
00:45:13 --> 00:45:19
It means
if I throw up this golf ball--

733
00:45:15 --> 00:45:21
I throw it up like this--

734
00:45:17 --> 00:45:23
and it has a certain component
in x direction

735
00:45:19 --> 00:45:25
a certain velocity

736
00:45:21 --> 00:45:27
if I move myself with
exactly that same velocity--

737
00:45:24 --> 00:45:30
with exactly the same
horizontal velocity--

738
00:45:27 --> 00:45:33
I could catch the ball here.

739
00:45:28 --> 00:45:34
It would have to come back
exactly in my hands.

740
00:45:31 --> 00:45:37
That is because there is only an
acceleration in the y direction

741
00:45:36 --> 00:45:42
but the motion
in the y direction

742
00:45:38 --> 00:45:44
is completely independent
of the x direction.

743
00:45:41 --> 00:45:47
The x direction
doesn't even know

744
00:45:43 --> 00:45:49
what's going on
with the y direction.

745
00:45:45 --> 00:45:51
In the x direction,
if I throw an object like this

746
00:45:48 --> 00:45:54
the x direction simply,
very boringly,

747
00:45:51 --> 00:45:57
moves with a constant velocity.

748
00:45:54 --> 00:46:00
There is no time dependence.

749
00:45:57 --> 00:46:03
And the y direction,
on its own, does its own thing.

750
00:46:00 --> 00:46:06
It goes up, comes to a halt
and it stops.

751
00:46:04 --> 00:46:10
And, of course, the
actual motion is the sum

752
00:46:07 --> 00:46:13
the superposition of the two.

753
00:46:11 --> 00:46:17
We have tried to find a way

754
00:46:13 --> 00:46:19
to demonstrate
this quite bizarre behavior

755
00:46:18 --> 00:46:24
which is not so intuitive.

756
00:46:20 --> 00:46:26
That the x direction
really lives a life of its own.

757
00:46:24 --> 00:46:30
And the way we want
to do that is as follows.

758
00:46:31 --> 00:46:37
We have here a golf ball

759
00:46:34 --> 00:46:40
a gun we can shoot
up the golf ball

760
00:46:39 --> 00:46:45
and we do that in such a way

761
00:46:41 --> 00:46:47
that the golf ball,
if we do it correctly

762
00:46:45 --> 00:46:51
exactly comes back here.

763
00:46:48 --> 00:46:54
That's not easy-- that takes
hours and hours of adjustments.

764
00:46:52 --> 00:46:58
The golf ball goes up
and comes back here.

765
00:46:57 --> 00:47:03
Not here, not here,
not there-- that's easy.

766
00:47:00 --> 00:47:06
You can shoot it up
a little at an angle

767
00:47:02 --> 00:47:08
and the golf ball
will come back here.

768
00:47:05 --> 00:47:11
Once we have achieved that--

769
00:47:07 --> 00:47:13
that the golf ball
will come back there--

770
00:47:09 --> 00:47:15
then I'm going to give
this cart a push

771
00:47:13 --> 00:47:19
and the moment that it passes
through this switch

772
00:47:17 --> 00:47:23
the golf ball will fire

773
00:47:19 --> 00:47:25
so that the golf ball
will go straight up

774
00:47:22 --> 00:47:28
as seen from the cart

775
00:47:24 --> 00:47:30
but it has a horizontal velocity

776
00:47:26 --> 00:47:32
which is exactly the same
horizontal velocity as the cart

777
00:47:29 --> 00:47:35
so the cart are like my hands.

778
00:47:31 --> 00:47:37
As the golf ball goes like this

779
00:47:33 --> 00:47:39
the cart stays always exactly
under the golf ball

780
00:47:37 --> 00:47:43
always exactly
under the golf ball

781
00:47:39 --> 00:47:45
and if all works well

782
00:47:41 --> 00:47:47
the ball ends up exactly
on the cart again.

783
00:47:46 --> 00:47:52
Let me first show you--

784
00:47:47 --> 00:47:53
otherwise, if that doesn't work,
of course, it's all over--

785
00:47:51 --> 00:47:57
that if we shoot the ball
straight up

786
00:47:53 --> 00:47:59
that it comes back here.

787
00:47:54 --> 00:48:00
If it doesn't do that

788
00:47:56 --> 00:48:02
I don't even have to try this
more complicated experiment.

789
00:48:01 --> 00:48:07
So here's the golf ball.

790
00:48:03 --> 00:48:09
I'm going to fire the gun now.

791
00:48:04 --> 00:48:10

792
00:48:07 --> 00:48:13
Close... close.

793
00:48:12 --> 00:48:18
Reasonably close.

794
00:48:15 --> 00:48:21
Well, since it's only
reasonably close, perhaps...

795
00:48:19 --> 00:48:25
(class laughs )

796
00:48:24 --> 00:48:30
Perhaps it would help if we
give it a little bit of leeway.

797
00:48:28 --> 00:48:34
There goes the gun.

798
00:48:33 --> 00:48:39
Here comes the ball.

799
00:48:38 --> 00:48:44
And this is just in case.

800
00:48:46 --> 00:48:52
Tape it down.

801
00:48:48 --> 00:48:54
So as I'm going to push
this now, give it a push

802
00:48:54 --> 00:49:00
the gun will be triggered

803
00:48:57 --> 00:49:03
when the middle
of the car is here.

804
00:49:00 --> 00:49:06
You've seen how high
that ball goes

805
00:49:01 --> 00:49:07
so that ball will go...
(makes whooshing sound )

806
00:49:05 --> 00:49:11
And depending upon
how hard I push it

807
00:49:07 --> 00:49:13
they may meet here
or they may meet there.

808
00:49:14 --> 00:49:20
You ready for this?

809
00:49:16 --> 00:49:22
You ready?

810
00:49:17 --> 00:49:23
CLASS:
Ready.

811
00:49:18 --> 00:49:24
LEWIN:
I'm ready.

812
00:49:19 --> 00:49:25

813
00:49:23 --> 00:49:29
Physics works.

814
00:49:24 --> 00:49:30
(class applauds )

815
00:49:28 --> 00:49:34
LEWIN:
See you Wednesday.

816
00:49:29 --> 00:49:35.000