TITLE: prove the following: $a⋅(b\times c)=(a\times b)⋅c$
QUESTION [0 upvotes]: Prove the following:
$a⋅(b\times c)=(a\times b)⋅c$
all letters are vectors, the x is cross product, the ⋅ is dot product

REPLY [1 votes]: My favorite formulation of this is that if $a,b,c$ are row vectors, then
$$a \cdot (b \times c) = \det \left( \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right) $$
This is particularly easy to see by comparing to the mnemonic for the cross product
$$ b \times c = \det \left( \begin{matrix}
\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}  
\\
b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right) $$
By permuting the rows around, we get the matrix computing $c \cdot (a \times b) = (a \times b) \cdot c$.
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