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Prove an identity relating scalar triple products of vectors A,B,C

by RoRi

Prove the following identity holds for vectors A,B,C \in \mathbb{R}^3.

    \[ (A \times B) \cdot (B \times C) \times (C \times A) = (A \cdot B \times C)^2. \]

Proof. From a previous exercise (Section 13.14, Exercise #9(d)), we know

    \[ (A \times B) \cdot (C \times D) = (B \cdot D)(A \cdot C) - (B \cdot C)(A \cdot D). \]

So, with B \times C in place of C and C \times A in place of D in this formula we have

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

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