1.4 The Cross Product
The second form of multiplication vector math defines is the cross product. Unlike the dot product, which evaluates to a scalar, the cross product evaluates to
another vector. Taking the cross product of two 3D vectors, and , yields another vector, , that is mutually orthogonal to and . By that we mean
is orthogonal to , and is orthogonal to ; see Figure 1.9. If = (u
x
, u
y
, u
z
) and = (v
x
,v
y
, v
z
), then the cross product is computed like so:
(1.5)
Figure 1.9: The cross product of two 3D vectors, and , yields another vector, , that is mutually orthogonal to and . If you take your left
hand and aim the fingers in the direction of the first vector , and then curl your fingers toward along an angle 0 = θ =
p
, then your thumb roughly
points in the direction of ; this is called the left-hand-thumb rule.
Example 1.6
Let = (2, 1, 3) and = (2, 0, 0). Compute and , and then verify that is orthogonal to and that is orthogonal
to . Applying formula 1.5 we have,
and
This result makes one thing clear, generally speaking: Therefore, we say that the cross product is anti-commutative. In fact, it can be
shown that . You can determine the vector returned by the cross product by the left-hand-thumb rule. If you curve the fingers of your
left hand from the direction of the first vector toward the second vector (always take the path with the smallest angle), your thumb points in the direction of the
returned vector, as shown in Figure 1.9.
To show that is orthogonal to and that is orthogonal to , we recall from §1.3 that if , then (i.e., the vectors are
orthogonal). Because
and
we conclude that is orthogonal to and that is orthogonal to .
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