FANDOM


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} $