What is the value of $(\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B})$?

$A$ vector $\vec{A}$ points vertically upward and $\vec{B}$ points towards north. The vector product $\vec{A} \times \vec{B}$ is

Two vectors $\vec{A}$ and $\vec{B}$ are at right angles to each other,when

Two adjacent sides of a parallelogram are represented by the two vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $3\hat{i} - 2\hat{j} + \hat{k}$. What is the area of the parallelogram?

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 $\vec{A} \perp \vec{B}$.

Vectors $a\hat{i} + b\hat{j} + \hat{k}$ and $2\hat{i} - 3\hat{j} + 4\hat{k}$ are perpendicular to each other. Given that $3a + 2b = 7$, and the ratio of $a$ to $b$ is $x/2$, find the value of $x$.