Let $\vec{a}, \vec{b}, \vec{c}$ be pair wise mutually perpendicular vectors, such that $|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=2$. Then length of $\vec{a}+\vec{b}+\vec{c}$ is equal to:

3

$|\vec{a}+\vec{b}+\vec{c}|^2$

$=(\vec{a}+\vec{b}+\vec{c}) .(\vec{a}+\vec{b}+\vec{c})$

$=|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2 . \vec{a} . \vec{b}+2 \vec{b} . \vec{c}+2 \vec{c} . \vec{a}$

= 1 + 4 + 4 + 0 + 0 + 0

= 9

$\Rightarrow |\vec{a}+\vec{b}+\vec{c}|=3$

Hence (3) is correct answer.