The Cross Product

The cross product (or vector product) is a binary operation on two vectors in a three-dimensional space.  The result is a vector which is perpendicular to both of the vectors being multiplied.

The cross product of two vectors a,b is denoted a x b (and sometimes u ^ v )

a x b = |a| |b| sin(a,b)

Cross vector

Properties of the dot product: 

1)u x v = - v x u

2)u x (v + w) = u x v + u x w

3)cu x v = c(u x v), where c is a scalar

4) u x ( v x w) = (u.w) v - (u.v) w and (u x v) x w = (u.w) v - (v.w) u

We can use the dot product along with the above properties to derive many important formulas and concepts in linear algebra. The proofs for many of these are found in the textbook, but here we will look just at the formulas and examples of their uses.