The planes of two rigid discs are perpendicular to each other. They are rotating about their axes. If their angular velocities are $3 \, rad/s$ and $4 \, rad/s$ respectively,then the resultant angular velocity of the system would be ........ $rad/s$.

Two vectors $\vec{A}$ and $\vec{B}$ are defined as $\vec{A} = a \hat{i}$ and $\vec{B} = a(\cos \omega t \hat{i} + \sin \omega t \hat{j})$,where $a$ is a constant and $\omega = \pi / 6 \text{ rad s}^{-1}$. If $|\vec{A} + \vec{B}| = \sqrt{3}|\vec{A} - \vec{B}|$ at time $t = \tau$ for the first time,the value of $\tau$,in seconds,is . . . . . .

The magnitude of the sum of the two vectors $\vec{A}$ and $\vec{B}$ is equal to the magnitude of the difference of the two vectors $\vec{A}$ and $\vec{B}$. The angle between $\vec{A}$ and $\vec{B}$ is: (in $^{\circ}$)

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\vec{A}$. What is the angle between $\vec{A}$ and $\vec{B}$?

Two forces,$F_1$ and $F_2$,are acting on a body. One force is double that of the other force and the resultant is equal to the greater force. Then the angle between the two forces is

Three vectors are expressed as $\vec{a} = 4\hat{i} - \hat{j}$,$\vec{b} = -3\hat{i} + 2\hat{j}$,and $\vec{c} = -\hat{k}$. The unit vector along the direction of the sum of these vectors is: