If the vectors $a\,i - 2j + 3k$ and $3i + 6j - 5k$ are perpendicular to each other,then $a$ is given by

$M$ and $N$ are the midpoints of the sides $BC$ and $CD$ of a parallelogram $ABCD$ respectively,then $\overline{AM} + \overline{AN} =$

The value of $x$ for which the angle between the vectors $a = -3i + xj + k$ and $b = xi + 2xj + k$ is acute and the angle between $b$ and the $x$-axis lies between $\pi/2$ and $\pi$ satisfies:

$ABCD$ is a parallelogram. The position vectors of $A$ and $C$ are respectively $3\hat{i} + 3\hat{j} + 5\hat{k}$ and $\hat{i} - 5\hat{j} - 5\hat{k}$. If $M$ is the midpoint of the diagonal $DB$,then the magnitude of the projection of $\vec{OM}$ on $\vec{OC}$,where $O$ is the origin,is

If two vectors $\vec{a}$ and $\vec{b}$ which are perpendicular to each other are such that $|\vec{a}|=8$ and $|\vec{b}|=3$, then $|\vec{a}-2\vec{b}|=$

Let $a = i + 2j + k$, $b = i - j + k$, and $c = i + j - k$. $A$ vector in the plane of $a$ and $b$ has a projection of $\frac{1}{\sqrt{3}}$ on $c$. Then, one such vector is: