The cross (or vector) product of the vectors a and b is defined as

a×b=|a||b| sinqn

where q is the angle between a and b, and n is the unit vector perpendicular to both a and b. The direction of n is that which a right-handed corkscrew would move when turned from a to b.

If a=(a₁,a₂,a₃) and b=(b₁,b₂,b₃) then we can also define the cross product as

a×b=(a₂ b₃ - a₃ b₂, a₃ b₁ - a₁ b₃, a₁ b₂ - a₂ b₁)

(this is the determinant of matrix, but due to my poor formatting skills I couldn't post it clearly - see here for more details).

For example, (3,0,-2)×(0,1,3)=(2,-9,3).

To answer the last part of your question, the cross product is useful if you want to find a vector perpendicular to a and b.

Hope this helps,