If $P(2, 3, 1)$ and $Q(7, 15, 1)$,then $|\overrightarrow{PQ}| = \dots$

If $G$ is the centroid of the $\triangle ABC$,then $\vec{GA} + \vec{GB} + \vec{GC}$ is equal to

$A$ vector which is in the direction of $(3, 6, 2)$ and has magnitude $4$ is $.......$

If the magnitude of the sum of two unit vectors is greater than the magnitude of their difference and less than $\sqrt{3}$ times the magnitude of their difference,then the complete set of values for the angle $\theta$ between the vectors is

Compute the magnitude of the following vectors:
$\vec{a}=\hat{i}+\hat{j}+\hat{k} ; \quad \vec{b}=2 \hat{i}-7 \hat{j}-3 \hat{k} ; \quad \vec{c}=\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}$

If $\vec{a} = \hat{i} + \hat{j} - 2 \hat{k}$,$\vec{b} = 2 \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = 3 \hat{i} - \hat{k}$. If $\vec{c} = m \vec{a} + n \vec{b}$ then $m + n = $