If $a, b, c$ are mutually perpendicular unit vectors,then $|a + b + c| = $

Two adjacent sides of a parallelogram $PQRS$ are given by $\vec{PQ} = \hat{i} + \hat{k}$ and $\vec{PS} = \hat{i} - \hat{j}$. If the side $PS$ is rotated about the point $P$ by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side $PQ$, then $\sin^2(\frac{5\alpha}{2}) - \sin^2(\frac{\alpha}{2})$ is equal to:

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - 3\hat{j} + 2\hat{k}$ and $\vec{c} = 3\hat{i} - 2\hat{k}$. If a vector $\vec{p}$ satisfies the conditions $\vec{p} \cdot \vec{c} = 0$ and $\vec{p} \times \vec{a} = \vec{b} \times \vec{a}$, then the value of $|\vec{p}|$ is:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors and $\vec{a}+\vec{b}+\vec{c}=\vec{0}$,then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is . . . . . . . (in $/2$)

If $\theta$ is the angle between the unit vectors $\mathbf{a}$ and $\mathbf{b}$,then $\cos \frac{\theta}{2} = $

Let $\vec{b}$ and $\vec{c}$ be non-collinear vectors satisfying $\vec{a} \times (\vec{b} \times \vec{c}) + (\vec{a} \cdot \vec{b})\vec{b} = (4 - 2x - \sin y)\vec{b} + (x^2 - 1)\vec{c}$ and $(\vec{c} \cdot \vec{c})\vec{a} = \vec{c}$,then $x$ is equal to