If $\overline{OA} = 3\hat{i} + \hat{j} - \hat{k}$,$|\overline{AB}| = 2\sqrt{6}$ and the direction ratios of $\overline{AB}$ are $1, -1, 2$,then $|\overline{OB}| = $

$A$ vector of magnitude $14$ lies in the $xy-$ plane and makes an angle of $60^\circ$ with the $x-$ axis. The components of the vector in the direction of the $x-$ axis and $y-$ axis are:

$a$ and $b$ are non-collinear vectors. If $c=(x-2)a+b$ and $d=(2x+1)a-b$ are collinear vectors,then the value of $x = \ldots$

If a parallelogram is constructed on the vectors $\vec{a} = 3\vec{p} - \vec{q}$ and $\vec{b} = \vec{p} + 3\vec{q}$, where $|\vec{p}| = 3$, $|\vec{q}| = 2$ and the angle between $\vec{p}$ and $\vec{q}$ is $\pi/3$, then the ratio of the lengths of adjacent sides $|\vec{a}|$ and $|\vec{b}|$ of the parallelogram is:

If the points $P(4, 5, x)$,$Q(3, y, 4)$ and $R(5, 8, 0)$ are collinear,then the value of $x + y$ is

The unit vector in the direction of $\vec{x} = (2, 3, \sqrt{3})$ is . . . . . . .