$A$ person moves $30 \,m$ North and then $20 \,m$ towards East and finally $30 \sqrt{2} \,m$ in South-West direction. The displacement of the person from the origin will be

Two forces of magnitude $F$ have a resultant of the same magnitude $F$. The angle between the two forces is ........ $^o$

Two forces whose magnitudes are in the ratio $5:3$ are acting at a point at an angle $60^{\circ}$ simultaneously. If the resultant of the two forces is $35 \ N$,then the magnitudes of the two forces respectively are

When are the magnitudes of $\vec{A} + \vec{B}$ and $\vec{A} - \vec{B}$ equal?

The resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ has a magnitude $R_{1}$. If the direction of $\overrightarrow{Q}$ is reversed,the resultant has a magnitude $R_{2}$. The value of $(R_{1}^{2} + R_{2}^{2})$ is:

Magnitudes of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are $4$ units and $3$ units respectively. If these vectors are $(i)$ in the same direction $(\theta = 0^{\circ})$ and $(ii)$ in the opposite direction $(\theta = 180^{\circ})$,find the magnitude of the resultant vector in each case.