Let $a, b$ and $c$ be non-zero vectors such that $(a \times b) \times c = \frac{1}{3}|b||c|a$. If $\theta$ is the acute angle between the vectors $b$ and $c$,then $\sin \theta$ equals

Let $a, b$ and $c$ be three unit vectors such that $a \times (b \times c) = \frac{1}{\sqrt{2}}(b + c)$ and $b$ is not parallel to $c$. If $\alpha$ and $\beta$ are the angles between $a, b$ and $a, c$ respectively,then $\alpha - \beta =$

If $\bar{x} \cdot \bar{y} = 0$,then $\bar{x} \times (\bar{x} \times \bar{y}) = \dots$ where $|\bar{x}| = 1$.

Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$,$\vec{b}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c}=5$,then $|\vec{c}|$ is equal to

Given three unit vectors $a, b, c$ such that $a \perp b$ and $a \parallel c$,then $a \times (b \times c)$ is

If $(\bar{a} \times \bar{b}) \times \bar{c} = -5 \bar{a} + 4 \bar{b}$ and $\bar{a} \cdot \bar{b} = 3$,then the value of $\bar{a} \times (\bar{b} \times \bar{c})$ is