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

$\vec A$ 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 ......... $^\circ$.

If a vector $2\hat i + 3\hat j + 8\hat k$ is perpendicular to the vector $4\hat j - 4\hat i + \alpha \hat k$,then the value of $\alpha$ is:

The two vectors have magnitudes $3$ and $5$. If the angle between them is $60^o$,then the dot product of the two vectors will be:

The diagonals of a parallelogram are $2\,\hat{i}$ and $2\,\hat{j}$. What is the area of the parallelogram in square units?

If two vectors $A$ and $B$ are mutually perpendicular,then the component of $A-B$ along the direction of $A+B$ is