If $a, b, c$ are three vectors such that $a = b + c$ and the angle between $b$ and $c$ is $\pi / 2$,then:

Let $a$ and $b$ be two unit vectors inclined at an angle $\theta$,then $\sin(\theta/2)$ is equal to

Let $(x, y) \in (R \times R)$ and $\vec{a} = x \hat{i} + 2 \hat{j} - \hat{k}$,$\vec{b} = 6 \hat{i} - y \hat{j} + 2 \hat{k}$ be two vectors. If $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = f(x) g(y)$,then $f(x) + g(y) - 46 = 0$ represents:

If the angles between the sides of the triangle $ABC$ formed by $A(2,3,5)$,$B(-1,3,2)$ and $C(3,5,-2)$ are $\alpha, \beta$ and $\gamma$,then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = $

Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. $A$ point $P$ moves so that at any time $t$ the position vector $\overline{OP}$ (where $O$ is the origin) is given by $\hat{a} \cos t + \hat{b} \sin t$. When $P$ is farthest from origin $O$,let $M$ be the length of $\overline{OP}$ and $\hat{u}$ be the unit vector along $\overline{OP}$. Then,

The vector $2\hat{i} + a\hat{j} + \hat{k}$ is perpendicular to the vector $2\hat{i} - \hat{j} - \hat{k},$ if $a = $