If $a$,$b$ and $c$ are non-coplanar vectors and $d$ is a unit vector, then find the value of
$$ \| (a \cdot d) ( b \times c) + (b \cdot d) (c \times a) + (c \cdot d) ( a \times b) \| $$
independent of d.
My approach Let $ A= a \times ((b \times c) \times d)$
$$\begin{aligned} A & = (a \cdot d) (b \times c) - ( a \cdot ( b \times c) ) d \\ & = (a \cdot d) ( b \times c) - ([a\,b\,c]) (d) \end{aligned}$$
After this step i am confused.
Let $V=$ the expression inside the modulus signs. First, using the BAC CAB rule, we can calculate $$\underline{d}\times\underline{V}$$
I don't wish to write it out in full, but it is easy to check that this is zero.
So $$\underline{V}=\lambda\underline{d}$$
However, writing $\Delta=\underline{a}\cdot(\underline{b}\times\underline{c})$, we have$$\underline{V}\cdot\underline{a}=(\underline{a}\cdot\underline{d})\Delta=\lambda(\underline{a}\cdot\underline{d})$$
So $\lambda=\Delta$ and then $$|\underline{V}|=\Delta=|\underline{a}\cdot(\underline{b}\times\underline{c})|$$