For any vector $\vec{a} \in \mathbb{R}^3$,$|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2 = $ . . . . . . .

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$ and $\vec{a} \times \vec{c} = \vec{b}$, $\vec{a} \cdot \vec{c} = 3$, then $\vec{c}$ is

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors. Suppose $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{6}$. Then $\vec{a}$ is

Let $\overline{a}=\hat{i}+2 \hat{j}-\hat{k}$ and $\overline{b}=\hat{i}+\hat{j}-\hat{k}$ be two vectors. If $\overline{c}$ is a vector such that $\overline{b} \times \overline{c}=\overline{b} \times \overline{a}$ and $\overline{c} \cdot \overline{a}=0$,then $\overline{c} \cdot \overline{b}$ is

Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2\hat{j}+3\hat{k}$. If the position vectors of the points $A, B$ and $C$ are $-2\hat{i}+\hat{j}-3\hat{k}$,$2\hat{i}+4\hat{j}-2\hat{k}$ and $-4\hat{i}+2\hat{j}-\hat{k}$ respectively,then the projection of the vector $\overline{OP}$ on a vector perpendicular to the vectors $\overline{AB}$ and $\overline{AC}$ is $......$.

Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three unit vectors,out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\vec{b}$ and $\vec{c}$ respectively and $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2} \vec{b},$ then $|\alpha - \beta|$ is equal to .............. $^o$