The position vector of a moving particle at time $t$ is $r = 3 \hat{i} + 4t \hat{j} - t \hat{k}$. Its displacement during the time interval $t = 1 \ s$ to $t = 3 \ s$ is

Why is a frame of reference required?

Explain path length.

$A$ balloon is moving up in air vertically above a point $A$ on the ground. When it is at a height $h_{1}$,a girl standing at a distance $d$ (point $B$) from $A$ sees it at an angle $45^{\circ}$ with respect to the vertical. When the balloon climbs up a further height $h_{2}$,it is seen at an angle $60^{\circ}$ with respect to the vertical if the girl moves further by a distance $2.464d$ (point $C$). Then the height $h_{2}$ is (given $\tan 30^{\circ} = 0.5774$):

The magnitude of the displacement is equal to the distance covered in a given interval of time if the particle

Explain with an example that the magnitude of displacement may be zero but the corresponding path length is not zero.