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We now would like to
generalize our concept of work

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to the motion of
an object that goes

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in more than one dimension.

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And so, if we have
our object here.

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And we're applying a force.

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And the object is moving.

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And let's say the path is a
little bit more complicated.

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So our object, a little
bit later, has moved.

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And we're going to
call that displacement.

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We now want to
generalize our concept

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to work to handle
this type of motion

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in more than one dimension.

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In order to do that, we'll need
a new mathematical operation,

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which we're going to
call the scalar product.

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So what I'd like to
do first, is define

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this in terms of
vectors and then

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we'll apply it to
the concept of work.

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So suppose I have two vectors,
a, and another vector, b.

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And they're separated
by an angle, theta.

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And in this case,
our theta, we're

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going to just take
it between 0 and pi.

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Now, I'd like to talk about
how much one vector projects

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in the direction of the other.

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So if I call this the
parallel component of b.

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Then I want to define a quantity
which we're going to call

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the scalar product of a dot b.

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Which is the
magnitude of a times

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the amount of b that is in
the direction parallel to a.

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From our geometric
diagram, you can

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see that b parallel is
equal to the magnitude of b

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times the cosine of theta.

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Remember, theta's
going from 0 to pi.

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So this quantity,
b parallel, can

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both be positive or negative.

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If theta is between 0 and
pi/2, then in that range,

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the cosine goes from 1 to 0.

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And this quantity, b, will
be greater or equal to 0.

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b parallel.

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If theta is going from
pi/2, theta to pi,

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then we have that b parallel
is less than or equal to 0.

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And, in particular,
at the value pi/2,

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b parallel is 0 because
when theta is pi/2,

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the b vector is completely
perpendicular to a.

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And this is what
we're going to define

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to be the scalar product.

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But geometrically, we can
also look at-- let's draw

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our pictures again-- a and b.

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Let's consider how much
of the a is parallel to b.

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So that's a parallel.

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And if this is the
angle theta, we

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see that a parallel--
and I'll write it

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as a parallel-- is the magnitude
of a times cosine of theta.

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And so, I can also
define a dot b

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as equal to how much of
how much of a is parallel

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to b times the magnitude of b.

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In both of these
instances, because b

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parallel is the magnitude
of b cosine theta,

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and a parallel is a magnitude
of a cosine theta, what we're

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writing is that a dot b is the
magnitude of a cosine theta

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magnitude of b.

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And now you can see that
this quantity is a parallel.

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And this quantity is b parallel.

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And this is our geometric
definition of a dot product.

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One thing that we
want to consider

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are two important
rules for dot products.

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And they're the following.

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That if we take a vector and
multiply a by a quantity-- c,

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here, is a scalar
quantity-- then

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this is equal to
c times a dot b.

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Geometrically, this
is very easy to see.

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Because if we draw
a and we draw b.

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And if I multiply a by a
scalar, and let's make,

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in this particular
picture, our scalar

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bigger than 1, then the
new vector a is multiplied

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by c and it's c times a.

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The scalar product is just ca
dot b, is the magnitude of ca

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times how much of b is parallel.

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And when you multiply
this together,

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you get c times a
times to b parallel.

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c of a dot b is equal to c
times the magnitude of a times

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b parallel.

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And you can see that these
two expressions are equal.

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If you multiply a dot
product by a scalar,

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it satisfies this rule.

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And there's one
other crucial rule

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that we'll need when
we look at vectors.

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That if you take a vector a
and you add to it, a vector b,

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and you dotted in
a vector c, then

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this is vector addition
and multiplication.

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This vector addition distributes
over vector multiplication

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by a dot c plus b dot c.

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So these two facts are crucial
to our development of vectors.

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One thing that we
can say is-- we'll

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leave this as a little exercise
for you to prove this result.

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It's a pretty straightforward
vector construction.

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And to just give you a
little bit of a hint.

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If we have a vector a
and another vector b

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and we have a similar
vector c, then

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if I draw the vector a
dot b plus b and I want

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to take a dot c.

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So I have how much of
a is parallel to c.

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And if I look at how much
of b is parallel to c, then

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as part of the exercise,
make sure that you

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can see that how much of
a plus b parallel to c

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agrees to this
distributive rule.

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So these are our definition
of scalar product.

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And the two key facts that
we'll need when we apply it

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in our example of work.