Given two vectors $\vec{A} = 3\hat{i} + \hat{j}$ and $\vec{B} = \hat{j} + 2\hat{k}$. If these two vectors represent the adjacent sides of a parallelogram,find the area of the parallelogram.

If $|\hat{a} \cdot \hat{b}| = \frac{1}{2}$,then $|\hat{a} - \hat{b}|$ may be $:-$

If $\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors,then which of the following are correct?
$(a) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp \overrightarrow{A}$
$(b) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp \overrightarrow{B}$
$(c) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp (\overrightarrow{A} + \overrightarrow{B})$
$(d) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp (\overrightarrow{A} - \overrightarrow{B})$
$(e) \ (\overrightarrow{A} \times \overrightarrow{B}) \perp (\overrightarrow{A} \cdot \overrightarrow{B})$

$\vec A$ and $\vec B$ are two vectors and $\theta$ is the angle between them. If $|\vec A \times \vec B| = \sqrt{3}(\vec A \cdot \vec B)$,the value of $\theta$ is ......... $^\circ$.

The angle between the two vectors $(2\hat{i} + 3\hat{j} + \hat{k})$ and $(-3\hat{i} + 6\hat{k})$ is ...... $^\circ$.

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