The magnitude of the vector resulting from the addition of two vectors,$6\hat{i} + 7\hat{j}$ and $3\hat{i} + 4\hat{j}$,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.

Let $\overrightarrow C = \overrightarrow A + \overrightarrow B$. Which of the following statements are correct?
$(A)$ It is possible to have $|\overrightarrow C| < |\overrightarrow A|$ and $|\overrightarrow C| < |\overrightarrow B|$.
$(B)$ $|\overrightarrow C|$ is always greater than $|\overrightarrow A|$.
$(C)$ $|\overrightarrow C|$ may be equal to $|\overrightarrow A| + |\overrightarrow B|$.
$(D)$ $|\overrightarrow C|$ is never equal to $|\overrightarrow A| + |\overrightarrow B|$.

$A$ vector $P$ directed along the $X$-axis is added to vector $Q$ which has a magnitude of $10 \ m$. The resultant vector is directed along the $Y$-axis,with a magnitude that is $2$ times that of $P$. The magnitude of $P$ is

The resultant of two forces acting at an angle of $120^{\circ}$ is $10 \,kg$-wt and is perpendicular to one of the forces. That force 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: