Definition:Right-Hand Rule/Cross Product
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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:
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 $\mathbf a \times \mathbf b$.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $4$. The Vector Product