$$\displaystyle (\vec{a}\times \vec{b})\times \vec{c}=\frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$$

.

根据二重外积公式:

$$\displaystyle (\vec{a}\cdot \vec{c})\vec{b}-(\vec{b}\cdot \vec{c})\vec{a}=\frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$$

$$\displaystyle \Rightarrow (\vec{a}\cdot \vec{c})\vec{b}-(\vec{b}\cdot \vec{c}+\frac{1}{3}|\vec{b}||\vec{c}|)\vec{a}=0$$

因为$\vec{a},\vec{b},\vec{c}$不共线.

$$\displaystyle \Rightarrow (\vec{a}\cdot \vec{c})=0,(\vec{b}\cdot \vec{c}+\frac{1}{3}|\vec{b}||\vec{c}|)=0$$

下面利用点积的定义,可以得到

$$|\vec{b}|\cdot |\vec{c}|cos\theta +\frac{1}{3}|\vec{b}||\vec{c}|=0$$

$$\displaystyle \Rightarrow cos\theta =-\frac{1}{3}$$

$$\displaystyle \Rightarrow sin\theta =\sqrt{1-cos^2\theta }=\frac{2\sqrt{2}}{3}$$