How do I find the cross product of two vectors and what does its direction mean?

The cross product a × b gives a new vector perpendicular to both a and b. Compute it as a 3x3 determinant with the first row i, j, k, the second row the components of a, and the third row the components of b. Expand along the top row, remembering the minus sign on the j term. Its magnitude is |a||b| sin θ, which equals the area of the parallelogram formed by the two vectors. The direction is given by the right hand rule: point your right hand fingers from a toward b, and your thumb points along a × b. Note a × b = −(b × a), so order matters. If two vectors are parallel, sin θ = 0 and the cross product is the zero vector, a handy test for parallel vectors.