Let $\hat{a}$ and $\hat{b}$ be unit vectors. If $\vec{c}$ is a vector such that the angle between $\hat{a}$ and $\vec{c}$ is $\frac{\pi}{12}$,and $\hat{b} = \vec{c} + 2(\vec{c} \times \hat{a})$,then $|6\vec{c}|^{2}$ is equal to

If the vectors $\bar{a}=\hat{i}-\hat{j}+2\hat{k}$,$\bar{b}=2\hat{i}+4\hat{j}+\hat{k}$ and $\bar{c}=m\hat{i}+\hat{j}+n\hat{k}$ are mutually perpendicular,then $(m, n)$ is

Let $ABC$ be an acute scalene triangle,and $O$ and $H$ be its circumcentre and orthocentre respectively. Further,let $N$ be the mid-point of $OH$. The value of the vector sum $\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$ is

Let $\overrightarrow{x}$ be a vector in the plane containing vectors $\overrightarrow{a} = 2\hat{i} - \hat{j} + \hat{k}$ and $\overrightarrow{b} = \hat{i} + 2\hat{j} - \hat{k}$. If the vector $\overrightarrow{x}$ is perpendicular to $(3\hat{i} + 2\hat{j} - \hat{k})$ and its projection on $\overrightarrow{a}$ is $\frac{17\sqrt{6}}{2}$,then the value of $|\overrightarrow{x}|^{2}$ is equal to ...... .

If $\vec{a}$,$\vec{b}$,and $\vec{a}-\vec{b}$ are unit vectors and the angle between the two vectors $\vec{a}$ and $\vec{b}$ is $\theta$,then $\theta = $ . . . . . . .

If $\vec{a}$ and $\vec{b}$ are two vectors such that $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} < 0$ and $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$,then the angle between the vectors $\vec{a}$ and $\vec{b}$ is