The cross-product \(\bm{u}\times\bm{v}\) will be a vector perpendicular (orthogonal) to both \(\bm{u}\) and \(\bm{v}.\) For \(\bm{u} \langle a,b,c, \rangle\) and \(\bm{v} = \langle p,q,r \rangle\) it is computed as \[\bm{u}\times\bm{v} = \operatorname{det}\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a & b & c \\ p & q & r \end{pmatrix} = \operatorname{det}\begin{pmatrix}b & c \\ q & r\end{pmatrix}\mathbf{i} - \operatorname{det}\begin{pmatrix}a & c \\ p & r\end{pmatrix}\mathbf{j} + \operatorname{det}\begin{pmatrix}a & b \\ p & q\end{pmatrix}\mathbf{k} = \big(br-cq\big)\mathbf{i} - \big(ar-cp\big)\mathbf{j} + \big(aq-bp\big)\mathbf{k} \]
Determinant of order two
and determinant of order three
We can calculate a vector perpendicular (normal) to a plane.
right-hand rule
Two vectors \(\bm{u}\) and \(\bm{v}\) are parallel if and only if \(\bm{u} \times \bm{v} = \bm{0}.\) More generally, \[ |\bm{u} \times \bm{v}| = |\bm{u}||\bm{v}|\sin(\theta)\,, \] which equals the area of the parallelogram defined by \(\bm{u}\) and \(\bm{v}\).
The scalar triple product of vectors \(\bm{u}\) and \(\bm{v}\) and \(\bm{w}\) is \[ \bm{u} \cdot (\bm{v} \times \bm{w} = \operatorname{det}\begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{pmatrix} \] and gives the volume of the parallelepiped SPANNED by those three vectors. Additionally, we can tell those vectors all lie in the same plane (are coplanar) if their scalar triple product is zero.