If vectors $\vec{A} = (2, -3, 1)$ and $\vec{B} = (3, 4, n)$ are perpendicular to each other,find the value of $n$.

Explain the cross product of two vectors.

The angle between the two vectors $\overrightarrow{A} = 5\hat{i} + 5\hat{j}$ and $\overrightarrow{B} = 5\hat{i} - 5\hat{j}$ will be ....... $^\circ$.

If $\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$,find the angle between $\vec{A}$ and $\vec{B}$.

For what value of $x$ will the two vectors $\vec{A} = 2\hat{i} + 2\hat{j} - x\hat{k}$ and $\vec{B} = 2\hat{i} - \hat{j} - 3\hat{k}$ be perpendicular to each other?

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