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Prove a formula for the cross product of vectors in terms of the unit coordinate vectors

Prove that we have the formula

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


Proof. Let A = (a_1, a_2, a_3) and B = (b_1, b_2, b_3). Then we compute,

    \begin{align*}  A &\cdot (B \times \mathbf{i}) \mathbf{i} + A \cdot (B \times \mathbf{j}) \mathbf{j} + A \cdot (B \times \mathbf{k}) \mathbf{k} \\  &= A \cdot (0, b_3, -b_2) \mathbf{i} + A \cdot (-b_3, 0, b_1) \mathbf{j} + A \cdot (b_2, -b_1, 0) \mathbf{k} \\  &= (a_2 b_3 - a_3 b_2) \mathbf{i} + (a_3 b_1 - a_1 b_3) \mathbf{j} + (a_1 b_2 - a_2 b_1) \mathbf{k} \\  &= A \times B. \qquad \blacksquare \end{align*}

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