Definition:Vector Cross Product

Let $$\vec {u}$$ and $$\vec v$$ be $3$-dimensional vectors, such that:
 * $$\vec u = u_i \mathbf{i} + u_j \mathbf{j} + u_k \mathbf{k}$$
 * $$\vec v = v_i \mathbf{i} + v_j \mathbf{j} + v_k \mathbf{k}$$

Then the vector cross product, denoted $$\vec u \times \vec v$$, is defined as:
 * $$\vec u \times \vec v = \begin{vmatrix}

\mathbf{i} & \mathbf{j} & \mathbf{k}\\ u_i & u_j & u_k \\ v_i & v_j & v_k \\ \end{vmatrix}$$

where $$\begin{vmatrix} \ldots \end{vmatrix}$$ can be understood as a determinant.

More directly, $$\vec u \times \vec v = (u_j v_k - u_k v_j)\mathbf{i} - (u_i v_k - u_k v_i)\mathbf{j} + (u_i v_j - u_j v_i)\mathbf{k}$$