= Four-vector scalar triple product identity
{title2=$[b,c,d]a-[a,c,d]b+[a,b,d]c-[a,b,c]d=0$}
For four <vectors> in three-dimensional real <Euclidean space>, write $[a,b,c]=a\cdot(b\times c)$. Expand the same <cross product> in two ways using the <vector triple product identity>:
$$
(a\times b)\times(c\times d)
=[a,b,d]c-[a,b,c]d
=[a,c,d]b-[b,c,d]a.
$$
Rearrangement gives the displayed identity. It expresses <linear dependence> of four <vectors> in three <dimensions> with explicit <determinant> coefficients and remains valid when some or all of those coefficients vanish.
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