Past exam of the mathematical methods I course of the University of Oxford / 2019 / 8 / Solution

Apply the vector triple product identity $p\times (c\times d)=c(p\cdot d)-d(p\cdot c)$ with $p=a\times b$. This givesFor column vectors $a,b,d$, the scalar triple product is $(a\times b)\cdot d=\det (a,b,d)$; replacing $d$ by $c$ gives the other coefficient. ThereforeThe signs follow the stated column convention and the cyclic invariance of the scalar triple product. No independence assumptions on the four vectors are needed.