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Prove or disprove a given vector formula

by RoRi

Prove or disprove:

    \[ A \times \big( A \times (A \times B) \big) \cdot C = -\lVert A \rVert^2 A \cdot B \times C. \]

Proof. We can compute using the identities we have derived in the previous exercises of this section,

    \begin{align*} A \times \big( A \times (A \times B) \big) &= A \times \big( (B \cdot A) A - (A \cdot A) B \big) \cdot C \\ &= \big( (A \times A)(B \cdot A) - \lVert A \rVert^2 (A \times B) big) \cdot C \\ &= -\lVert A \rVert^2 ((A \times B) \cdot C) \\ &= -\lVert A \rVert^2 ((A \cdot B) \times C). \qquad \blacksquare \end{align*}

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