Let a, b and c be three non-zero vectors such that no two of them are collinear and a( b c…

Question

Let $\vec a,\vec b$ and $\vec c$ be three non-zero vectors such that no two of them are collinear and $\vec a	imes(\vec b	imes\vec c)=\dfrac13|\vec b||\vec c|\vec a$. If $	heta$ is the angle between vectors $\vec b$ and $\vec c$, then a value of $\sin	heta$ is:

Accepted Answer

$$\vec a\times(\vec b\times\vec c) = -\vec c\times(\vec a\times\vec b) = -(\vec c\cdot\vec b)\vec a+(\vec c\cdot\vec a)\vec b = \frac13|\vec b||\vec c|\vec a$$

Since $\vec a,\vec b,\vec c$ are non-collinear, this forces $\vec c\cdot\vec a=0$ and $-|\vec b||\vec c|\cos\theta=\dfrac13|\vec b||\vec c|$, i.e. $\cos\theta=-\dfrac13$.

$$\sin\theta = \sqrt{1-\frac19}=\frac{2\sqrt2}3$$

Solutions · 1