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:

The value of $\hat{i} \times (\hat{i} \times \vec{a}) + \hat{j} \times (\hat{j} \times \vec{a}) + \hat{k} \times (\hat{k} \times \vec{a})$ is

What will be the projection of vector $\vec{A} = \hat{i} + \hat{j} + \hat{k}$ on vector $\vec{B} = \hat{i} + \hat{j}$?

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

Explain the cross product of two vectors.

For any two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$,if $\overrightarrow{A} \cdot \overrightarrow{B} = |\overrightarrow{A} \times \overrightarrow{B}|$,the magnitude of $\overrightarrow{C} = \overrightarrow{A} + \overrightarrow{B}$ is equal to