Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}$ and $\bar{c}=\hat{j}-\hat{k}$ be three vectors such that $\bar{a} \times \bar{b}=\bar{c}$ and $\bar{a} \cdot \bar{c}=0$. If the length of the projection vector of the vector $\bar{b}$ on the vector $\bar{a} \times \bar{c}$ is $l$,then the value of $3l^2$ is

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

If $a, b, c$ are three unit vectors such that $a \times (b \times c) = \frac{\sqrt{3}}{2} b + \frac{1}{2} c$,then the angles between $a, b$ and $a, c$ respectively are

Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k}$,$\vec{a} \cdot \vec{b}=1$ and $\vec{a} \times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6 \vec{b}$ is equal to

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 $a, b$ and $c$ are non-zero vectors such that $a \times b = c$ and $b \times c = a$,then $a \times c$ is