Cross product

The cross product is one way of taking the product of two vectors (the other being the dot product). This method yields a third vector perpendicular to both. It is defined by the formula
:\mathbf{a} \times \mathbf{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin \theta \ \mathbf{n}
where \mathbf{n} is the unit vector perpendicular to both \mathbf{a} and \mathbf{b}. It can be computed other ways as well:
:\mathbf{a} \times \mathbf{b} = \begin{vmatrix}
\mathbf{i}&\mathbf{j}&\mathbf{k}\\
a_x & a_y & a_z\\
b_x & b_y & b_z\\
\end{vmatrix} =
\begin{vmatrix}
a_y & a_z\\
b_y & b_z
\end{vmatrix}\mathbf{i}
-\begin{vmatrix}
a_x & a_z\\
b_x & b_z
\end{vmatrix}\mathbf{j}
+\begin{vmatrix}
a_x & a_y\\
a_x & a_y
\end{vmatrix}\mathbf{k}
:= (a_y b_z - a_z b_y)\mathbf{i}+(a_z b_x - a_x b_z)\mathbf{j}+(a_x b_y - a_y b_x)\mathbf{k}
Category:Mathematical operations