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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.