Let the vectors be ( mathbf{a} = langle a_1, a_2, a_3 rangle ) and ( mathbf{b} = langle b_1, b_2, b_3 rangle )
Write the cross product as ( mathbf{a} times mathbf{b} )
Use the determinant form:
( mathbf{a} times mathbf{b} = begin{vmatrix} mathbf{i} & mathbf{j} & mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 end{vmatrix} )
Expand the determinant:
( mathbf{a} times mathbf{b} = mathbf{i}(a_2 b_3 – a_3 b_2) – mathbf{j}(a_1 b_3 – a_3 b_1) + mathbf{k}(a_1 b_2 – a_2 b_1) )
Write the result in component form:
( mathbf{a} times mathbf{b} = langle a_2 b_3 – a_3 b_2,; a_3 b_1 – a_1 b_3,; a_1 b_2 – a_2 b_1 rangle )
Compute each component separately
Use the right-hand rule for direction
Reverse the order to get the negative:
( mathbf{b} times mathbf{a} = -(mathbf{a} times mathbf{b}) )
Check if the vectors are parallel:
Parallel vectors give ( mathbf{a} times mathbf{b} = mathbf{0} )