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NCERT Class 12 · Ch 11 Exercise 11.2 · Q2

NCERT Class 12 Maths · Chapter 11 Three Dimensional Geometry · Exercise 11.2 Question 2

Show that the directions of the line through (1,−1,2),(3,4,−2) and the line through (0,3,2),(3,5,6) make a right angle.

💡 Concept Hints

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

✅ Step-by-Step Solution

Final Answer: Their directions are orthogonal; the spatial lines themselves are skew.
  1. Their directions are d=(2,5,−4) and e=(3,2,4).
  2. d·e=6+10−16=0, so the angle between their directions is π/2.
  3. Equating points (1,−1,2)+td=(0,3,2)+se gives t=14/11,s=13/11 from x,y, but these fail z. Thus the lines do not intersect.

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