Compute the magnitude of the following vectors:

The given vectors are:

$\vec{a}=\hat{i}+\hat{j}+\hat{k} ; \quad \vec{b}=2 \hat{i}-7 \hat{j}-3 \hat{k}$$\vec{c}=\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}$

$|\vec{a}|=\sqrt{(1)^{2}+(1)^{2}+(1)^{2}}=\sqrt{3}$

$|\vec{b}|=\sqrt{(2)^{2}+(-7)^{2}+(-3)^{2}}$

$=\sqrt{4+49+9}$

$=\sqrt{62}$

$|\vec{c}|=\sqrt{\left(\frac{1}{\sqrt{3}}\right)^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}+\left(-\frac{1}{\sqrt{3}}\right)^{2}}$

$=\sqrt{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=1$