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 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 $\vec P$

A displacement vector of magnitude $4$ makes an angle $30^{\circ}$ with the $x$-axis. Its rectangular components in $x-y$ plane are .........

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  ........... $N$

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\overrightarrow{\mathrm{A}}$ and its magnitude is half that of $\vec{B}$. The angle between vectors $\vec{A}$ and $\vec{B}$ is . . . . . . 

The values of $x$ and $y$ for which vectors $\vec A = \left( {6\hat i + x\hat j - 2\hat k} \right)$ and $\vec B = \left( {5\hat i - 6\hat j - y\hat k} \right)$ may be parallel are