Vector triple product

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Why is it generally true that $\mathbf{a}\times(\mathbf{b}\times\mathbf{c}) \neq (\mathbf{a}\times\mathbf{b})\times\mathbf{c}$ ?

A.The cross product is not associative — the two sides expand to different combinations of dot products

B.The cross product is commutative, so order never matters

C.Both sides are always zero, so the question is meaningless

D.Vector multiplication is always associative, so they actually are always equal

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If $\mathbf{a}$ is perpendicular to both $\mathbf{b}$ and $\mathbf{c}$ , what is $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})$ ?

\mathbf{0}

\mathbf{b}\times\mathbf{c}

C.A vector of magnitude

|\mathbf{a}||\mathbf{b}||\mathbf{c}|

along

\mathbf{a}

D.It cannot be determined without more information

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For $\mathbf{a}=2\mathbf{i}+2\mathbf{j}-\mathbf{k}$ and $\mathbf{b}=\mathbf{c}=3\mathbf{i}-6\mathbf{j}+2\mathbf{k}$ (i.e. $\mathbf{b}=\mathbf{c}$ ), what is $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})$ ?

\mathbf{0}

, since

\mathbf{b}\times\mathbf{c}=\mathbf{0}

when

\mathbf{b}=\mathbf{c}

B.A nonzero vector along

\mathbf{a}

C.A nonzero vector along

\mathbf{b}

D.It is undefined since

\mathbf{b}

and

\mathbf{c}

are identical

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Vector triple product

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