8 ROTATIONAL MOTION 8.4 The vector product×××× ×a x b (^) |a x b| = a b sin
b a
Figure 70: The vector product.
Is it also possible to combine two vector multiplicatively to form a third (non-
coplanar) vector? It turns out that this goal can be achieved via the use of the
so-called vector product. By definition, the vector product, a b, of two vectors
a = (ax, ay, az) and b = (bx, by, bz) is of magnitude
|a × b| = |a| |b| sin θ. (8.12)
The direction of a b is mutually perpendicular to a and b, in the sense given by
the right-hand grip rule when vector a is rotated onto vector b (the direction of
rotation being such that the angle of rotation is less than 180 ◦). See Fig. 70. In
coordinate form,
a × b = (ay bz − az by, az bx − ax bz, ax by − ay bx). (8.13)
There are a number of fairly obvious consequences of the above definition.
Firstly, if vector b is parallel to vector a, so that we can write b = λ a, then the
vector product a b has zero magnitude. The easiest way of seeing this is to note
that if a and b are parallel then the angle θ subtended between them is zero,
hence the magnitude of the vector product, |a| |b| sin θ, must also be zero (since
sin 0 ◦ = 0). Secondly, the order of multiplication matters. Thus, b × a is not
equivalent to a × b. In fact, as can be seen from Eq. (8.13),
b × a = −a × b. (8.14)
In other words, b a has the same magnitude as a b, but points in diagram-
matically the opposite direction.