The angle between two vectors $-2\hat{i} + 3\hat{j} + \hat{k}$ and $\hat{i} + 2\hat{j} - 4\hat{k}$ is ....... $^o$

If $\vec A$ and $\vec B$ are perpendicular vectors,where $\vec A = 5\hat i + 7\hat j - 3\hat k$ and $\vec B = 2\hat i + 2\hat j - a\hat k$,then the value of $a$ is:

If $\overrightarrow{P} = 3\hat{i} + \sqrt{3}\hat{j} + 2\hat{k}$ and $\overrightarrow{Q} = 4\hat{i} + \sqrt{3}\hat{j} + 2.5\hat{k}$,then the unit vector in the direction of $\overrightarrow{P} \times \overrightarrow{Q}$ is $\frac{1}{x}(\sqrt{3}\hat{i} + \hat{j} - 2\sqrt{3}\hat{k})$. The value of $x$ is:

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.

What is the angle between the resultant of $\overrightarrow{A} + \overrightarrow{B}$ and $\overrightarrow{A} \times \overrightarrow{B}$?

If the vector $(\hat{a} + 2\hat{b})$ is perpendicular to the vector $(5\hat{a} - 4\hat{b})$,then the angle between $\hat{a}$ and $\hat{b}$ is ........ $^\circ$.