If $\vec{a}$ and $\vec{b}$ are vectors in space given by $\vec{a}=\frac{\hat{i}-2 \hat{j}}{\sqrt{5}}$ and $\vec{b}=\frac{2 \hat{i}+\hat{j}+3 \hat{k}}{\sqrt{14}}$,then the value of $(2 \vec{a}+\vec{b}) \cdot[(\vec{a} \times \vec{b}) \times(\vec{a}-2 \vec{b})]$ is

$a \times (b \times c) + b \times (c \times a) + c \times (a \times b) =$

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b} \times(\vec{b} \times \vec{c}) .$ If magnitudes of the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are $\sqrt{2}, 1$ and $2$ respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta$ $(0 < \theta < \frac{\pi}{2})$,then the value of $1+\tan \theta$ is equal to:

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

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.

The vectors $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying $\vec{b} \times \vec{c} = \vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 0$. Then the vector $\vec{d}$ is equal to: