The magnitude of the $X$ and $Y$ components of $\vec{A}$ are $7$ and $6$ respectively. Also,the magnitude of the $X$ and $Y$ components of $\vec{A} + \vec{B}$ are $11$ and $9$ respectively. What is the magnitude of $\vec{B}$?

Statement $I:$ Two forces $(\overrightarrow{P}+\overrightarrow{Q})$ and $(\overrightarrow{P}-\overrightarrow{Q})$,where $\overrightarrow{P} \perp \overrightarrow{Q}$,act at an angle $\theta_{1}$ to each other,and the magnitude of their resultant is $\sqrt{3(P^{2}+Q^{2})}$. When they act at an angle $\theta_{2}$,the magnitude of their resultant becomes $\sqrt{2(P^{2}+Q^{2})}$. This is possible only when $\theta_{1} < \theta_{2}$.
Statement $II:$ In the situation given above,$\theta_{1} = 60^{\circ}$ and $\theta_{2} = 90^{\circ}$.
In the light of the above statements,choose the most appropriate answer from the options given below.

The vectors $5\hat{i} + 8\hat{j}$ and $2\hat{i} + 7\hat{j}$ are added. The magnitude of the sum of these vectors is

$A$ particle undergoes three successive displacements given by $s_1 = \sqrt{2} \ m$ north-east,$s_2 = 2 \ m$ due south,and $s_3 = 4 \ m$ at $30^{\circ}$ north of west. The magnitude of the net displacement is:

Three vectors each of magnitude $3 \sqrt{1.5}$ units are acting at a point. If the angle between any two vectors is $\frac{\pi}{3}$,then the magnitude of the resultant vector of the three vectors is

If the direction of a vector $\overrightarrow{A}$ is reversed,find $\Delta \overrightarrow{A}$ and $|\Delta \overrightarrow{A}|$.