If a vector $A=2i+2j+3k$ and $B=3i+6j+nk$ are perpendicular to each other,then the value of $n$ is

Two unit vectors $\hat{a}_{1}$ and $\hat{a}_{2}$ are inclined to each other at an angle $\theta$. If $|\hat{a}_{1}-\hat{a}_{2}|=\sqrt{3}$,then the value of $(\hat{a}_{1}-\hat{a}_{2}) \cdot (2\hat{a}_{1}-\hat{a}_{2})$ is:

Write the necessary condition for the scalar product of two mutually perpendicular vectors.

Define the vector product of two vectors.

If $\vec{P} = (k, 2, 3)$ and $\vec{Q} = (0, 3, k)$ and $\vec{P} \perp \vec{Q}$,then what is the value of $k$?

If $|\vec A \times \vec B| = \sqrt 3 \vec A \cdot \vec B,$ then the value of $|\vec A + \vec B|$ is