If $\bar{a}=\frac{1}{\sqrt{10}}(3 \hat{\imath}+\hat{k})$ and $\bar{b}=\frac{1}{7}(2 \hat{\imath}+3 \hat{\jmath}-6 \hat{k})$,then the value of $(2 \bar{a}-\bar{b}) \cdot[(\bar{a} \times \bar{b}) \times(\bar{a}+2 \bar{b})]$ is

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

If $\vec{a} = -\hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + 0\hat{j} + \hat{k}$,find a vector $\vec{c}$ satisfying the following conditions:
$(i)$ $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$.
$(ii)$ $\vec{c}$ is perpendicular to $\vec{b}$.
$(iii)$ $\vec{a} \cdot \vec{c} = 7$.

$\overrightarrow{a} \times [\overrightarrow{a} \times (\overrightarrow{a} \times \overrightarrow{b})]$ is equal to

If $a, b, c, d$ are coplanar vectors,then $(a \times b) \times (c \times d) = $

Let $\overline{a}=\hat{j}-\hat{k}$ and $\overline{c}=\hat{i}-\hat{j}-\hat{k}$. Then the vector $\overline{b}$ satisfying $\overline{a} \times \overline{b}+\overline{c}=\overline{0}$ and $\overline{a} \cdot \overline{b}=3$,is