If $\overrightarrow{A} \times \overrightarrow{B} = \overrightarrow{B} \times \overrightarrow{A}$,then the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is

If $\hat{A}$ is a unit vector in a given direction,then the value of $\hat{A} \cdot \frac{d\hat{A}}{dt}$ is

For three vectors $\vec{A} = (-x \hat{i} - 6 \hat{j} - 2 \hat{k})$,$\vec{B} = (-\hat{i} + 4 \hat{j} + 3 \hat{k})$ and $\vec{C} = (-8 \hat{i} - \hat{j} + 3 \hat{k})$,if $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$,then the value of $x$ is:

An ant starts from the origin and crawls $10 \ cm$ along the $x$-axis and then $20 \ cm$ along the $y$-axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45^{\circ}$ with the $x$-axis and has a magnitude of $\sqrt{2} \ cm$ is

Find the unit vector perpendicular to $\vec{A}$ and $\vec{B}$ where $\vec{A} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{B} = \hat{i} + 2\hat{j}$.

The angle between the vectors $(\hat{i} + \hat{j})$ and $(\hat{i} - \hat{k})$ is ........ $^\circ$.