3D Geometry
Distance between a skew line and a plane
MMTS_Full_Test_16
Grade 12
Question:
The distance between the line $\vec{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda(\hat{i}-\hat{j}+4\hat{k})$ and the plane $\vec{r}\cdot(\hat{i}+5\hat{j}+\hat{k})=5$ is
(A) $\dfrac{10}{9}$
(B) $\dfrac{10}{3\sqrt{3}}$
(C) $\dfrac{10}{3}$
(D) $\dfrac{5}{3}$
Step-by-Step Solution
Key Concept: First check that the line is parallel to the plane: $(\hat{i}-\hat{j}+4\hat{k})\cdot(\hat{i}+5\hat{j}+\hat{k})=1-5+4=0$ ✓. Then find distance from a point on the line to the plane.
Line parallel to plane. Distance $=\frac{|2-10+3-5|}{\sqrt{27}}=\frac{10}{3\sqrt{3}}$.
Correct Answer: (B) $\dfrac{10}{3\sqrt{3}}$