Vectors & 3D Geometry
Intersecting lines in 3D — parameter and distance match
MJAT_TS7_P1
Grade 12
Question:
Let $\gamma\in\mathbb{R}$ such that $L_1: \frac{x+1}{11}=\frac{y+2}{21}=\frac{z+3}{29}$ and $L_2: \frac{x+3}{16}=\frac{y+2}{11}=\frac{z+4}{\gamma}$ intersect at $R_1$. Match P)$\gamma$; Q)unit normal $\hat{n}$; R)$\overrightarrow{OR_1}$; S)$\overrightarrow{OR_1}\cdot\hat{n}$
with List-II: 1)$-\hat{i}-\hat{j}+\hat{k}$; 2)$\frac{3}{2}$; 3)$1$; 4)$\frac{1}{\sqrt{6}}\hat{i}-\frac{2}{\sqrt{6}}\hat{j}+\frac{1}{\sqrt{6}}\hat{k}$; 5)$\frac{2}{3}$
A) P→3, Q→4, R→1, S→2
B) P→5, Q→4, R→1, S→2
C) P→3, Q→4, R→1, S→5
D) P→3, Q→1, R→4, S→5
Step-by-Step Solution
Key Concept: For $L_1,L_2$ to intersect: set parametric equations equal and solve for $\gamma$. The normal to the plane containing both lines is $\vec{d_1}\times\vec{d_2}$. Compute $R_1$ and the projections.
P→(3), Q→(4), R→(1), S→(5). Answer: **C**.
Correct Answer: C