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

For unit vectors $\bar{a}, \bar{b}, \bar{c}$,if $\bar{a} \times (\bar{b} \times \bar{c}) = \frac{\bar{b}}{2}$ and $\bar{b}, \bar{c}$ are non-collinear vectors,then the angles made by $\bar{a}$ with $\bar{b}$ and $\bar{c}$ respectively are:

For any three vectors $a, b, c$,the condition $a \times (b \times c) = (a \times b) \times c$ holds if:

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

If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(\overline{a}-2 \overline{b}) \cdot \{(\overline{a} \times \overline{b}) \times (2 \overline{a}+\overline{b})\}$ is

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three non-zero vectors such that no two of them are collinear and $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3}|\overline{b}||\overline{c}| \overline{a}$. If $\theta$ is the angle between vectors $\overline{b}$ and $\overline{c}$,then the value of $\operatorname{cosec} \theta$ is