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NCERT Class 12 · Ch 11 Miscellaneous Exercise · Q5

NCERT Class 12 Maths · Chapter 11 Three Dimensional Geometry · Miscellaneous Exercise Question 5

Find the vector line through (1,2,−4) whose direction is orthogonal to (x−8)/3=(y+19)/(−16)=(z−10)/7 and (x−15)/3=(y−29)/8=(z−5)/(−5).

💡 Concept Hints

Identify the base point and a nonzero direction before applying a vector formula.
Use dot products for angles, proportional directions for parallelism, and a triple product for skew-line distance.

✅ Step-by-Step Solution

Final Answer: r=(1,2,-4)+t(2,3,6), t∈ℝ

The direction vectors are d=(3,-16,7) and e=(3,8,-5).
A direction perpendicular to both is d×e=(24,36,72), proportional to (2,3,6).
The required line is r=(1,2,-4)+t(2,3,6), t∈ℝ. Substitution gives zero dot product of its direction with both d and e.

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