proof for $[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$

Question

I encounter this triple product property in wikipedia But I can't find proof for $$[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$$
The RHS cross product produce vector while the LHS produce scalar.
So this got me stumble on working out this equation.
How do I get scalar equals to vector?
Does anyone know proof for this?

Answer 1

All quantities below are vectors. I will use the following properties of cross-products and dot-products:

$$ (x \times y) \times z = (x \cdot z) y - (y \cdot z)x \\ x \cdot ( y \times z) = y \cdot (z \times x) = z \cdot (x \times y) \\ x \cdot (x \times y) = 0 $$

We start with the righthand side. For convenience, denote $a \times c = v$. Then

\begin{align} (a \times b) \times (a \times c) = (a \times b) \times v &= (a \cdot v)b - (b \cdot v) a \\ &= (a \cdot(a \times c) )b - (b \cdot(a \times c)) a \\ &= 0 - (- a \cdot (b \times c))a \\ &= (a \cdot (b \times c) )a \end{align}