Cross product

The cross product is an operation that takes two nonzero vectors and produces a vector (not a scalar) perpendicular to both of them. Geometrically, we define the magnitude of the cross product with

|vec u xx vec v| = |vec u||vec v|sin theta,

where theta is the angle between the vectors in standard position. To find the direction of the vector, we use the right-hand rule: point in the direction of vec u with your fingers, and then curl them (naturally, not backwards) towards the direction of vec v. Your thumb will then point in the direction of vec u xx vec v.

Algebraically, we define the cross product in RR^3 with

[u_1,u_2,u_3] xx [v_1,v_2,v_3] = [w_1,w_2,w_3],

where the components are given by

Component Value Remember
w_1 u_2v_3 - u_3v_2 2, 3
w_2 u_3v_1 - u_1v_3 3, 1
w_3 u_1v_2 - u_2v_1 1, 2

The pattern shouldn’t be too hard to see. Each part has a product of two terms, and then you just subtract the same two terms with the subscripts swapped. You essentially need to remember the sequence 233112, and even that has some repetition. You do not need to remember anything for RR^2, since the cross product exists only in RR^3 and in RR^7.

Like the dot product, the cross product has some important properties. First, it can produce the zero vector:

vec v xx vec 0 = vec 0 qquad and qquad vec v xx vec v = vec 0.

Unlike the dot product, is is anticommutative—order matters:

vec u xx vec v = -(vec u xx vec v).

vec u xx (vec v + vec w) = vec u xx vec v + vec u xx vec w.

It is associative over scalar multiplication:

kvec u xx vec v = k(vec u xx vec v) = vec u xx kvec v.

We can use the dot product and cross product together to make a test for coplanarity. Here it is: vectors vec u, vec v, and vec w are coplanar if and only if

vec u * vec v xx vec w = 0.

It does not matter which vectors are placed where in that equation. As long as crossing two of the vectors and dotting the result with the third yields zero, they are coplanar. Why does this work? When we cross vec v and vec w, we get a vector perpendicular to both of them. If vec u is on the same plane, it should also be perpendicular to the cross product. We test for this using the dot product—two vectors are perpendicular if their dot product is zero.

The notation vec u * vec v xx vec w is unambiguous; it is the same as vec u * (vec v xx vec w). If you dotted first, you would then be crossing a scalar with a vector, and that is not possible.