Show that if $a,b,c,d\in{\mathbb R}^3$, then $d= $$\frac{ [(b\times c)\cdot d]a + [(c\times a)\cdot d]b + [(a\times b)\cdot d]c} { [(a\times b)\cdot c]} $
If $a=\left ( x_{1},x_{2},x_{3} \right )$, $b=\left ( y_{1},y_{2},y_{3} \right )$, $c=\left ( z_{1},z_{2},z_{3} \right )$ and $d=\left ( w_{1},w_{2},w_{3} \right )$, it is clear that the proposed exercise is demonstrated, my question is, if there is another way for your demonstration. Thanks for your help.
First show that it holds for the standard basis $a, b, c$ and every $d$.
Then show it still holds if any of $a, b, c$ is multiplied by a nonzero scalar.
Then show it still holds when e.g. $a$ is exchanged to $a+b$, and in all similar cases.
Finally, by these elementary operations, we can arrive to any $a, b, c$ with nonzero determinant.