If $a \cdot b = \beta$ and $a \times b = c$, then $a =$

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 $\vec{a}, \vec{b},$ and $\vec{c}$ are vectors such that $|\vec{b}| = |\vec{c}|$,then $[(\vec{a} + \vec{b}) \times (\vec{a} \times \vec{c})] \times (\vec{b} \times \vec{c}) \cdot (\vec{b} + \vec{c}) = ...$

If $\overline{b}$ and $\overline{c}$ are unit vectors and $|\bar{a}|=7$,$\bar{a} \times(\bar{b} \times \bar{c})+\bar{b} \times(\bar{c} \times \bar{a})=\frac{1}{2} \bar{a}$,then the angle between the vectors $\bar{a}$ and $\overline{c}$ and the angle between the vectors $\overline{b}$ and $\overline{c}$ are respectively.

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