The position vectors of two points $A$ and $B$ are $\hat{i} + \hat{j} - \hat{k}$ and $2\hat{i} - \hat{j} + \hat{k}$ respectively. Then $|\overrightarrow{AB}| = $

If the length of a vector is $21$ and its direction ratios are $2, -3, 6$,then its direction cosines are:

$\vec{c}$ is a unit vector in the direction of the sum of vectors $\vec{a}$ and $\vec{b}$. Where,$\vec{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}$,then $|\vec{c}| = $ . . . . . . .

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let $S$ be any point on the vector $\vec{R}$. The distance between the point $P$ and $S$ is $b|\vec{R}|$. The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are parallel vectors,then which of the following is true?

If $\vec{a} = \hat{i}+\hat{j}-\hat{k}$ and $\vec{b} = 2\hat{i}-3\hat{j}+\hat{k}$ are adjacent sides of a parallelogram,then the lengths of its diagonals are