Magnitudes of vectors $\vec{a}, \vec{b}, \vec{c}$ are $3, 4, 5$ respectively. If $\vec{a}$ and $\vec{b} + \vec{c}$,$\vec{b}$ and $\vec{c} + \vec{a}$,and $\vec{c}$ and $\vec{a} + \vec{b}$ are mutually perpendicular,then the magnitude of $\vec{a} + \vec{b} + \vec{c}$ is:

Two adjacent sides of a parallelogram are represented by $\vec{AB} = 2\hat{i} + 10\hat{j} + 11\hat{k}$ and $\vec{AD} = -\hat{i} + 2\hat{j} + 2\hat{k}$. The side $\vec{AD}$ is rotated by an acute angle $\theta$ in the plane of the parallelogram such that $\vec{AD}$ becomes $\vec{AD'}$. If $\vec{AD'}$ is perpendicular to $\vec{AB}$,find $\cos \theta$.

If $\hat{a}, \hat{b},$ and $\hat{c}$ are unit vectors satisfying $\hat{a} - \sqrt{3}\hat{b} + \hat{c} = \vec{0},$ then the angle between the vectors $\hat{a}$ and $\hat{c}$ is

The angle between the vectors $2 \hat{k} - 3 \hat{j}$ and $\hat{i} - 2 \hat{k}$ is

If $a$ and $b$ are respectively the internal and external bisectors of the angles between the vectors $u = -\hat{i} + 2\hat{j} - 2\hat{k}$ and $v = 3\hat{i} + 4\hat{j}$, and $|a| = \frac{2}{3}\sqrt{6}$, $|b| = \frac{2}{3}\sqrt{3}$, then one of the values of $a - b$ is

Let $\overline{A}=2 \hat{i}+\hat{k}$,$\overline{B}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{C}=4 \hat{i}-3 \hat{j}+7 \hat{k}$. If a vector $\overline{R}$ satisfies $\overline{R} \times \overline{B}=\overline{C} \times \overline{B}$ and $\overline{R} \cdot \overline{A}=0$,then $\overline{R}$ is given by