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scalar and a vector. The operation
a
(
b
×
c
) is known as the triple product.
We present some special properties of this computation in Section 6.1.
As mentioned earlier, the vector cross product is not commutative. In
fact, it is anticommutative:
a
×
b
= −(
b
×
a
). The cross product is not
associative, either. In general, (
a
×
b
) ×
c
=
a
× (
b
×
c
). More vector
algebra laws concerning the cross product are given in
Section 2.13.
2.12.2 Geometric Interpretation
The cross product yields a vector that is perpendicular to the original two
vectors, as illustrated in Figure 2.27.
Figure 2.27
Vector cross product
b
is equal to the product of the magnitudes of
a
and
b
and the sine of the angle between
a
and
b
:
The length of
a
×
a
×
b
=
a
b
sinθ.
The magnitude of the
cross product is related
to the sine of the angle
between the vectors
As it turns out, this is also equal to the area of the parallelogram formed
with two sides
a
and
b
. Let's see if we can verify why this is true by using
Figure 2.28.
Figure 2.28
A parallelogram with sides
a
and
b
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