Let the vectors be ( mathbf{a} = (a_1, a_2, a_3) ) and ( mathbf{b} = (b_1, b_2, b_3) )
Write 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} = (a_2 b_3 – a_3 b_2)mathbf{i} – (a_1 b_3 – a_3 b_1)mathbf{j} + (a_1 b_2 – a_2 b_1)mathbf{k} )
Write the result as a vector:
( mathbf{a} times mathbf{b} = (a_2 b_3 – a_3 b_2,; a_3 b_1 – a_1 b_3,; a_1 b_2 – a_2 b_1) )
Use the right-hand rule for direction
Swap the order to reverse the sign:
( mathbf{a} times mathbf{b} = -(mathbf{b} times mathbf{a}) )
If the vectors are parallel, the cross product is ( mathbf{0} )