Two forces $\vec{F}_1$ and $\vec{F}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{F}_1$ and $\vec{F}_2$ is $\cos^{-1}\left(\frac{1}{n}\right)$. The value of $|n|$ is . . . . . . .

The magnitudes of pairs of displacement vectors are given. Which pair of displacement vectors cannot be added to give a resultant vector of magnitude $13 \, cm$?

Given $A = \hat{i} + \hat{j} + \hat{k}$ and $B = -\hat{i} - \hat{j} - \hat{k}$,then the angle between $(A - B)$ and $A$ is: (in $^{\circ}$)

With respect to a rectangular cartesian coordinate system,three vectors are expressed as $\vec a = 4\hat i - \hat j$,$\vec b = -3\hat i + 2\hat j$,and $\vec c = -\hat k$,where $\hat i, \hat j, \hat k$ are unit vectors along the $X, Y,$ and $Z$-axis respectively. The unit vector $\hat r$ along the direction of the sum of these vectors is:

Vectors $\vec{A}$ and $\vec{B}$ make angles of $20^\circ$ and $110^\circ$ with the $X$-axis,respectively. The magnitudes of these vectors are $5 \, m$ and $12 \, m$,respectively. The magnitude of the resultant vector of these two vectors is ....... $m$.

If $\overrightarrow{R}$ is the resultant vector of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$,then $|\overrightarrow{R}|$ is . . . . . . $|\overrightarrow{A}| + |\overrightarrow{B}|$.