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 . . . . . . 

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

A metal sphere is hung by a string fixed to a wall. The sphere is pushed away from the wall by a stick. The forces acting on the sphere are shown in the second diagram. Which of the following statements is wrong

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

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 magnitude of the $X$ and $Y$ component of $\vec A$ are $7$ and $6$ respectively. Also the magnitude of $X$ and $Y$ component of $\vec A + \vec B$ are $11$ and $9$ respectively. What is the magnitude of $\vec B$  ?