How to Do a Cross Product?

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 determinant

( 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} )

Compute the ( mathbf{i} ) component: ( a_2 b_3 – a_3 b_2 )

Compute the ( mathbf{j} ) component: ( a_3 b_1 – a_1 b_3 )

Compute the ( mathbf{k} ) component: ( a_1 b_2 – a_2 b_1 )

Combine the components: ( 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 )

Apply the negative sign to the ( mathbf{j} ) component if using the determinant expansion form

Verify the result is perpendicular to both vectors

Use the right-hand rule to check direction