If $|\vec{a}|=2$ and $|\vec{b}|=3$,find the value of $|3 \vec{a}+2 \vec{b}|$ based on the given figure.

$A$ car travels $6 \, km$ at an angle of $45^o$ north of east and then travels a distance of $4 \, km$ at an angle of $135^o$ north of east. How far is the final point from the starting point? What angle does the straight line joining its initial and final position make with the east direction?

$A$ force ${F_1} = 1\,N$ acts along the direction of $x = 0$,and a force ${F_2} = 2\,N$ acts along the direction of $y = 0$. Find the resultant force.

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 equal forces ($P$ each) act at a point inclined to each other at an angle of $120^\circ$. The magnitude of their resultant is

$A$ vector $\vec{A}$ when added to the sum of the vectors $(\hat{i} - 2\hat{j} + 2\hat{k})$ and $(-2\hat{i} + \hat{j} - \hat{k})$ gives a unit vector along the $Y$-axis. The magnitude of vector $\vec{A}$ is