$\overrightarrow A \, = \,3\widehat i\, + \,2\widehat j$ , $\overrightarrow B \, = \widehat {\,i} + \widehat j - 2\widehat k$  then find their addition by algebric method.

The vector that must be added to the vector $\hat i - 3\hat j + 2\hat k$ and $3\hat i + 6\hat j - 7\hat k$ so that the resultant vector is a unit vector along the $y-$axis is

If vectors $P, Q$ and $R$ have magnitude $5, 12$ and $13 $ units and $\overrightarrow P + \overrightarrow Q = \overrightarrow R ,$ the angle between $Q$ and $R$ is

If the sum of two unit vectors is also a unit vector. then magnitude of their difference and angle between the two given unit vectors is ..............

If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $\mathrm{R}$ are inclined at an angle $\theta$, then

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