Definition:Right-Hand Rule/Cross Product

Definition
Let $\mathbf a$ and $\mathbf b$ be vector quantities.

The right-hand rule for the vector cross product $\mathbf a \times \mathbf b$ is a consequence of the determinant definition:
 * $\mathbf a \times \mathbf b = \begin{vmatrix}

\mathbf i & \mathbf j & \mathbf k\\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$

when embedded in the conventional right-hand Cartesian $3$-space:


 * Right-hand-3-space.png

Let a right hand be placed such that:
 * the thumb and index finger are at right-angles to each other
 * the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm
 * the thumb points along the direction of $\mathbf a$
 * the index finger points along the direction of $\mathbf b$.

Then the $3$rd finger is pointed along the direction of $\mathbf a \times \mathbf b$.


 * VectorCrossProduct.png