Any hints/solutions to how I can show
$$ [ \mathbf{a,b,c}] [ \mathbf{u,v,w}] = \begin{vmatrix} \mathbf{a.u} & \mathbf{a.v} & \mathbf{a.w} \\ \mathbf{b.u} & \mathbf{b.v} & \mathbf{b.w} \\ \mathbf{c.u} & \mathbf{c.v} & \mathbf{c.w} \\ \end{vmatrix}$$
I would like to avoid fiddly manipulation of individual components at all costs if possible. I noted its trivial if one of $\mathbf{a,b,c}$ or $\mathbf{u,v,w}$ are planar so I assumed u,v,w can be written as a linear combination of a,b,c but couldn't see the followup. Expanding the determinant just got a load of terms too so not sure how that could help. I suppose I have missed something obvious.
Any thoughts would be appreciated. (also I am new to this site so sorry about any problems with my question).
Hints
$1.$ For any two $3 \times 3$ matrix $A$ and $B$ we have
$$\text{Det}(AB)=\text{Det}(A)\text{Det}(B)$$
$2.$ For three $3 \times 1$ vectors ${\bf{a}}$, ${\bf{b}}$ and ${\bf{c}}$ constructing the columns of a $3 \times 3$ matrix $A$ we know that
$$[ \mathbf{a,b,c}]=\text{Det}(A)$$