= Reciprocal basis involution
{title2=$[\widehat e_1,\widehat e_2,\widehat e_3]=1/[e_1,e_2,e_3]$}
For a three-dimensional <basis> with nonzero <scalar triple product> $\Delta$, its <reciprocal basis> satisfies $\widehat e_2\times\widehat e_3=e_1/\Delta$ and the cyclic analogues. Consequently its <scalar triple product> is $1/\Delta$, and taking the <reciprocal basis> again recovers the original <basis>. The coordinates of $V$ in the original <basis> are $V\cdot\widehat e_i$; in the <reciprocal basis> they are $V\cdot e_i$.
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