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If we have three vectors: $n$,$a$, and $b$, and I want to evaluate $$\frac{(n \times a)\times(n \times b)}{2}$$, can I FOIL this to get:

$$\frac{(n \times n) + (n \times b) + (a \times n) + (a \times b)}{2}$$ = $$\frac{(n \times b) + (a \times n) + (a \times b)}{2}$$

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The vector triple product formula is $$a \times (b \times c)=(a\cdot c)b-(a\cdot b)c$$ where you need the parentheses because the cross product is not associative. If you apply this you get $$(n \times a)\times(n \times b)=((n\times a)\cdot b)n-((n \times a)\cdot n)b\\ =((n\times a)\cdot b)n$$ because the second dot product is zero.

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