$A$ vector $\vec{A}$ is rotated by a small angle $\Delta \theta$ radian $(\Delta \theta \ll 1)$ to get a new vector $\vec{B}$. In that case,$|\vec{B} - \vec{A}|$ is

The unit vector along $\hat{i} + \hat{j}$ is

For the resultant of two vectors $A$ and $B$ to be maximum,the angle between them should be (in $^{\circ}$)

The angle between vectors $\vec a$ and $\vec b$ is $\frac{\pi }{6}$. The angle between vectors $-3\vec a$ and $2\vec b$ is

The angles which a vector $\hat{i} + \hat{j} + \sqrt{2} \hat{k}$ makes with $X, Y$ and $Z$ axes respectively are

Two vectors $\overrightarrow{X}$ and $\overrightarrow{Y}$ have equal magnitude. The magnitude of $(\overrightarrow{X}-\overrightarrow{Y})$ is $n$ times the magnitude of $(\overrightarrow{X}+\overrightarrow{Y})$. The angle between $\overrightarrow{X}$ and $\overrightarrow{Y}$ is: