The vector product

Abstract

The vector product (often called the cross product) of two vectors a and b, denoted by a × b, or sometimes a ∧ b, is defined to be a vector of magnitude |a| |b| sin θ, where θ is the angle contained between a and b, with the direction of a × b being perpendicular to both a and b in the sense in which a right-handed corkscrew moves when rotated from a to b. Thus if n is a unit vector in this direction (see Fig. 6.1)

$$a\times b=\left( {\left| a \right|\left| b \right|\sin \theta } \right)n.$$

((6.1))

The ordered set of vectors {a, b, n} is said to be a right-handed set. (This is because of the way in which the direction of n is defined.) As {b, a, −n} is also a right-handed set of vectors, it follows that

$$a\times b=-b\times a$$

((6.2))

and so the vector product is not commutative.