Is the following relation correct? (vec a×vec b)×(vec a×vec…

Is the following relation correct? $(\vec a\times\vec b)\times(\vec a\times\vec c)\stackrel{?}{=}(\vec a\cdot\vec b)\,[(\vec a\times\vec b)\cdot\vec c]$.

Step 1: The left side

By Problem 144, or as in Problem 145: \[(\vec a\times\vec b)\times(\vec a\times\vec c)=\vec a\,\big[(\vec a\times\vec b)\cdot\vec c\big]-\vec c\,\big[(\vec a\times\vec b)\cdot\vec a\big]=[\vec a,\vec b,\vec c]\,\vec a.\]

Step 2: Verdict

The left side is a vector (along $\vec a$), but the proposed right side is a scalar. The relation is meaningless as written, and it fails in general. Counterexample with $\vec a=\vec i$, $\vec b=\vec j$, $\vec c=\vec k$:

\[ \vec k\times(-\vec j)=\vec i=\{1,0,0\},\qquad (\vec a\cdot\vec b)[\ldots]=0\cdot1=0. \]

\[\text{Incorrect. The correct identity is }(\vec a\times\vec b)\times(\vec a\times\vec c)=[\vec a,\vec b,\vec c]\,\vec a\]