Question Regarding Cross Product Identity

Question

I am trying to understand how $$\vec{r} \times(\alpha \times \vec{r}) = $$ $$\alpha(r \cdot r) - r(r \cdot \alpha)$$

The context of this question is shown in the image below. Thank you.

Answer 1

One interpretation: $a\times(b\times c)$ is orthogonal to $b\times c$, so it should be a linear combination of $b$ and $c$: $$a\times(b\times c) = ?b + ?c.$$The coefficient of $b$ should be formed by the remaining vectors, so follow your nose: this coefficient is $a\cdot c$. Similarly, you should expect the coefficient of $c$ to be $a\cdot b$, but the $\times$ operation is skew, so this gives a minus sign. So $$a\times(b\times c) = (a\cdot c)b - (a\cdot b)c.$$Make $a=c=r$ and $b = \alpha$ to get $$r\times (\alpha\times r) = |r|^2\alpha - (\alpha\cdot r)r.$$

Answer 2

This is a special case of the famous "BAC-CAB" rule. $$a\times (b\times c) = b(a\cdot c) - c(a\cdot b)$$