Answer
$\vec{a} \times \vec{b} = \det \begin{pmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix}$, resulting in a vector perpendicular to both $\vec{a}$ and $\vec{b}$.
Explanation
Magnitude is $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, representing the area of the parallelogram formed by the vectors.