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There's another
vector operation,

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which we call the vector product
and sometimes this operation

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is called the cross product.

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And this is taking two vectors
A, the operation cross,

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B. And this will
give a new vector C.

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And now we want to
define that new vector C.

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So let's draw a
three dimensional

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picture where we have a plane.

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And on this plane, we
have two vectors A and B.

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And those vectors are forming
an angle theta between them.

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Now a plane defines
two unit normals.

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I'll draw one up,
which I'm going

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to call n hat right hand rule.

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And the reason for
that right-hand rule

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is if we take A cross B,
then our right hand thumb

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is pointing in the perpendicular
direction to the plane.

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Now I could have chosen
a unit vector down,

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n hat left-hand rule.

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And this would correspond
to taking my left hand

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and having A cross
B pointing down.

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So the way we're going to
define our cross-product

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is with the right-hand rule.

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And so we define it like this.

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That the vector C has
magnitude, the magnitude

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of A times sine of theta,
the magnitude of B,

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and its direction is given
by the right-hand rule.

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Now one of the reasons
for this definition

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is let's draw a vector
A and a vector B.

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When you have two vectors,
they define a magnitude

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in the following way.

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That we can think
about any two vectors

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define an area of
a parallelogram.

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And we can define
that area as follows.

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Let's drop a perpendicular.

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And let's call this B perp.

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Then the area, which
is a positive quantity,

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is given by-- and by
the way, our angle theta

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here will be always
positive-- so we're

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going to make it 0 theta pi.

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And that way sine of theta is
always a positive quantity.

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The area is the height, so
that B perp times the base,

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which is the magnitude of A,
and so we can write that as A,

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and the magnitude of B
perp is B sine theta.

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So this quantity, B sine
theta, is precisely what's

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occurring there.

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So the magnitude of C is
equal to the area formed

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by the vectors A
and B. And we have

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a choice of which way we
want to pick C to point.

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And that's where,
as a convention,

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we're choosing the
right-hand rule.

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Now again, there
is a symmetry here

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in that I can also define
the area of that triangle

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in the following way.

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Let's write this, theta.

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Instead of taking
how much of B is

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perpendicular to
the direction of A,

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let's drop the
perpendicular this way,

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and write that as a perp.

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And then the same
area can be expressed

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as the magnitude of--
well, we'll express it

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as A perp times B. And
A perp is the magnitude

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of A sine theta times
the magnitude of B.

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And that's our same
definition as before.

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And so you see, a sine theta
is appearing over here.

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So in either choice,
our vector operation

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says take any two vectors.

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Any two vectors forms
a parallelogram.

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The area of that
parallelogram is the magnitude

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of the vector product.

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And the direction
of this new vector C

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is given by the right-hand
rule with respect

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to that parallelogram
in that sense.