$\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{k} \times \hat{i}) + \hat{k} \cdot (\hat{i} \times \hat{j}) = $

Let $\vec A = (\hat i + \hat j)$ and $\vec B = (2\hat i - \hat j)$. The magnitude of a coplanar vector $\vec C$ such that $\vec A \cdot \vec C = \vec B \cdot \vec C = \vec A \cdot \vec B$ is given by

The value of $(\overrightarrow A + \overrightarrow B ) \times (\overrightarrow A - \overrightarrow B )$ is

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 ........ $^\circ$.

Explain the kinds of multiplication operations for vectors.

If $\vec{A} = 4\hat{i} + n\hat{j} - 2\hat{k}$ and $\vec{B} = 2\hat{i} + 3\hat{j} + \hat{k}$,find the value of $n$ such that $\vec{A} \perp \vec{B}$.