The ratio of the maximum and minimum magnitudes of the resultant of two vectors $\vec{a}$ and $\vec{b}$ is $3 : 1$. Then $|\vec{a}|$ is equal to:

If the sum of two unit vectors is also a unit vector,then the magnitude of their difference and the angle between the two given unit vectors is ..............

Two forces of $12 \, N$ and $8 \, N$ act upon a body. The resultant force on the body has a maximum value of ........ $N$.

If the resultant of two forces has a magnitude smaller than the magnitude of the larger force,the two forces must be

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 displacement of a particle from a point having position vector $2 \hat{i} + 4 \hat{j}$ to another point having position vector $5 \hat{i} + 1 \hat{j}$ is ........ units.