The magnitude of pairs of displacement vectors are given. Which pair of displacement  vectors cannot be added to give a resultant vector of magnitude $13\, cm$?

A particle starting from the origin $(0,0)$ moves in a straight line in the $(x, y)$ plane. Its coordinates at a later time are $(\sqrt 3,3)$ . The path of the particle makes with the $x -$ axis an angle of  ....... $^o$

Three forces acting on a body are shown in the figure. To have the resultant force only along the $y-$ direction, the magnitude of the minimum additional force needed is  ........... $N$

A vector $\vec Q$ which has a magnitude of $8$ is added to the vector $\vec P$ which lies along $x-$ axis. The resultant of two vectors lies along $y-$ axis and has magnitude twice that of  $\vec P$. The magnitude of is $\vec P$

What is the maximum number of rectangular components into which a vector can be split in space?

The angles which a vector $\hat i + \hat j + \sqrt 2 \,\hat k$ makes with $X, Y$ and $Z$ axes respectively are