$\overrightarrow A \, = \,3\widehat i\, + \,2\widehat j$ , $\overrightarrow B \, = \widehat {\,i} + \widehat j – 2\widehat k$  then find their addition by algebric method.

A vector $\vec A $ is rotated by a small angle $\Delta \theta$ radian $( \Delta \theta << 1)$ to get a new vector $\vec B$ In that case $\left| {\vec B – \vec A} \right|$ is

If a particle moves from point $P (2,3,5)$ to point $Q (3,4,5)$. Its displacement vector be

Given $a+b+c+d=0,$ which of the following statements eare correct:

$(a)\;a, b,$ c, and $d$ must each be a null vector,

$(b)$ The magnitude of $(a+c)$ equals the magnitude of $(b+d)$

$(c)$ The magnitude of a can never be greater than the sum of the magnitudes of $b , c ,$ and $d$

$(d)$ $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of a and $d ,$ if they are collinear ?

On an open ground, a motorist follows a track that turns to his left by an angle of $60^{°}$ after every $500\; m$. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.