Which vector must be added to the resultant vector $\vec{P} = 2\hat{i} + 7\hat{j} - 10\hat{k}$ and $\vec{Q} = \hat{i} + 2\hat{j} + 3\hat{k}$ to obtain a unit vector along the $X$-axis?

Two forces $F_1 = 3 \ N$ at $0^o$ and $F_2 = 5 \ N$ at $60^o$ act on a body. Then a single force that would balance the two forces must have a magnitude of .......... $N$

The sum of two forces acting at a point is $16\, N.$ If the resultant force is $8\, N$ and its direction is perpendicular to the minimum force,then the forces are:

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ have equal magnitudes. If the magnitude of $\overrightarrow{A} + \overrightarrow{B}$ is equal to two times the magnitude of $\overrightarrow{A} - \overrightarrow{B}$,then the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ will be .......................

The sum of the magnitudes of two forces is $25 \,N$. The resultant of these forces is perpendicular to the smaller force and has a magnitude of $10 \,N$. The two forces are:

Give the equations to find the magnitude and direction of the resultant vector of two vectors $\vec{A}$ and $\vec{B}$ that are inclined at an angle $\theta$ to each other.