If $a, b, c$ are three vectors such that $a = b + c$ and the angle between $b$ and $c$ is $\pi / 2$,then:

Find the distance of the point $P(-\hat{i} + 2\hat{j} + 6\hat{k})$ from the line passing through the point $A(2\hat{i} + 3\hat{j} - 4\hat{k})$ and parallel to the vector $\vec{b} = 6\hat{i} + 3\hat{j} - 4\hat{k}$.

If $\triangle ABC$ is right-angled at $A$,where $A \equiv (4, 2, x)$,$B \equiv (3, 1, 8)$,and $C \equiv (2, -1, 2)$,then the value of $x$ is

Consider the following Assertion $(A)$ and Reason $(R)$:
Assertion $(A)$: The two lines $\bar{r}=\bar{a}+t(\bar{b})$ and $\bar{r}=\bar{b}+s(\bar{a})$ intersect each other.
Reason $(R)$: The shortest distance between the lines $\bar{r}=\bar{p}+t(\bar{q})$ and $\bar{r}=\bar{c}+s(\bar{d})$ is equal to the length of the projection of the vector $(\bar{p}-\bar{c})$ on $(\bar{q} \times \bar{d})$.
The correct answer is:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors and $\vec{a}+\vec{b}+\vec{c}=\vec{0}$,then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is . . . . . . . (in $/2$)

If $\overline{a}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\overline{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ are two vectors,then the angle between the vectors $3 \bar{a}+5 \bar{b}$ and $5 \bar{a}+3 \bar{b}$ is