3D Geometry
Shortest Distance — OM·ON
nta_pyq_2024_jan
Grade 12
Question:
Let O be the origin, and M and N be the points on the lines $\dfrac{x-5}{4}=\dfrac{y-4}{1}=\dfrac{z-5}{3}$ and $\dfrac{x+8}{12}=\dfrac{y+2}{5}=\dfrac{z+11}{9}$ respectively such that $MN$ is the shortest distance between the given lines. Then $\overrightarrow{OM}\cdot\overrightarrow{ON}$ is equal to
Step-by-Step Solution
Key Concept: Find points $M$ and $N$ on the two lines such that $MN\perp$ both direction vectors. Use the SD formula to find the parameter values, then compute $OM\cdot ON$.
$\overrightarrow{OM}\cdot\overrightarrow{ON}=9$.
Correct Answer: 9