The distance of the point $Q(0,2,-2)$ from the line passing through the point $P(5,-4,3)$ and perpendicular to the lines $\vec{r}=(-3\hat{i}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}+5\hat{k})$ and $\vec{r}=(\hat{i}-2\hat{j}+\hat{k})+\mu(-\hat{i}+3\hat{j}+2\hat{k})$ is
Step-by-Step Solution
Key Concept: Direction of required line = $(2,3,5)\times(-1,3,2)=(6-15,\,-5-4,\,6+3)=(-9,-9,9)\propto(-1,-1,1)$. Line through $P(5,-4,3)$ with direction $(-1,-1,1)$. Distance from $Q(0,2,-2)$ to this line.
Distance $=\sqrt{74}$.
Correct Answer: 4