Definition:Right-Hand Rule/Cross Product

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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:


Let a right hand be placed such that:

the $3$rd finger is at right-angles to both 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 $\mathbf a \times \mathbf b$.