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Prove an identity of cross products and the unit coordinate vectors

by RoRi

Prove that we have the identity:

    \[ \mathbf{i} \times (A \times \mathbf{i}) + \mathbf{j} \times (A \times \mathbf{j}) + \mathbf{k} \times (A \times \mathbf{k}) = 2A. \]

Proof. Let A = (a_1, a_2, a_3) and compute,

    \begin{align*} \mathbf{i} &\times (A \times \mathbf{i}) + \mathbf{j} \times (A \times \mathbf{j}) + \mathbf{k} (A \times \mathbf{k}) \\ &= \mathbf{i} \times (0, a_3, -a_2) + \mathbf{j} \times (-a_3, 0, a_1) + \mathbf{k} \times (a_2, -a_1, 0) \\ &= (0,a_2, a_3) + (a_1, 0, a_3) + (a_1, a_2, 0) \\ &= 2(a_1, a_2, a_3) \\ &= 2A. \end{align*}

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