Given three vectors $a=2 \hat{i}-\hat{j}+\hat{k}$,$b=\hat{i}+2 \hat{j}-\hat{k}$,and $c=\hat{i}+\hat{j}-2 \hat{k}$,a vector in the plane of $b$ and $c$ whose projection on $a$ is of magnitude $\sqrt{\frac{2}{3}}$ is

If $a, b, c$ and $r$ are vectors such that $a$ is not perpendicular to $b$,$r \times b = c \times b$ and $r \cdot a = 0$,then $r =$

Let $\overline{A}, \overline{B}, \overline{C}$ be vectors of lengths $3$ units,$4$ units,and $5$ units respectively. If $\overline{A}$ is perpendicular to $\overline{B}+\overline{C}$,$\overline{B}$ is perpendicular to $\overline{C}+\overline{A}$,and $\overline{C}$ is perpendicular to $\overline{A}+\overline{B}$,then the length of vector $\overline{A}+\overline{B}+\overline{C}$ is

In a trapezium $ABCD$,$\vec{BC} = \lambda \vec{AD}$ and $\vec{x} = \vec{AC} + \vec{BD}$. If $\vec{x} = p \vec{AD}$,then $p =$

Scalar projection of the line segment joining the points $A(-2,0,3)$ and $B(1,4,2)$ on the line whose direction ratios are $6,-2,3$ is

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,