Two vectors having equal magnitudes of $x$ units acting at an angle of $45^\circ$ have a resultant of $\sqrt{2 + \sqrt{2}}$ units. The value of $x$ is

The vector sum of two forces is perpendicular to their vector difference. In this case,the forces:

Two forces of magnitude $3\;N$ and $4\;N$ respectively are acting on a body. Calculate the resultant force if the angle between them is $0^{\circ}$. (in $;N$)

Give the equations to find the magnitude and direction of the resultant vector of two vectors $\vec{A}$ and $\vec{B}$ that are inclined at an angle $\theta$ to each other.

Two forces of $10 \,N$ and $6 \,N$ act upon a body. The directions of the forces are unknown. The resultant force on the body may be .........$N$.

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 . . . . . .