The perpendicular distance of point $(2,0,-3)$ from the line which passes through $(0,2,-4)$ and perpendicular to the lines $\vec{r}=(-3\hat{i}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}+5\hat{k})$ and $\vec{r}=(\hat{i}-2\hat{j}+\hat{k})+\mu(-\hat{i}+3\hat{j}+2\hat{k})$
Step-by-Step Solution
Key Concept: Direction of line = cross product of the two given direction vectors
Direction $=\vec{d_1}\times\vec{d_2}=(2,3,5)\times(-1,3,2)=(6-15,(-5-4),6+3)=(-9,-9,9)\sim(-1,-1,1)$. Distance of $(2,0,-3)$ from line through $(0,2,-4)$ with direction $(-1,-1,1)$: $=\frac{\sqrt{78}}{3}$.
Correct Answer: 2