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Rotational Invariance
Let's begin by defining the cross product for the unit vectors
i
,
j
, and
k
. Since all vectors can be
decomposed in terms of unit vectors (see Unit vectors), once we've defined the cross products for
this special case it will be easy to extend the definition to include all vectors. As we noted above,
the cross product between
i
and
j
(since they both lie in the
x

y
plane) must point purely in the
z
direction. Hence:
i
×
j
=
c
k
for some constant
c
. Because later on we will want the magnitude of the resultant vector to have
geometric significance, we need
c
k
to have unit length. In other words,
c
can be either +1 or 1.
Now we make a completely arbitrary choice in order to accord with convention: we choose
c
= +
1 . The fact that we have chosen
c
to be positive is known as The RightHand Rule (we could just
as easily have chosen
c
=  1 , and the math would all work out to be the same as long as we were
consistentbut we
do
have to choose one or the other, and there's no use going against what
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 Fall '10
 DavidJudd
 Physics

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