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NCERT Class 12 · Ch 11 Miscellaneous Exercise · Q1

NCERT Class 12 Maths · Chapter 11 Three Dimensional Geometry · Miscellaneous Exercise Question 1

Find the angle between nonzero directions (a,b,c) and (b−c,c−a,a−b). Assume a,b,c are not all equal, so both define lines.

💡 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: π/2.
  1. Their dot product is a(b−c)+b(c−a)+c(a−b)=ab−ac+bc−ab+ac−bc=0.
  2. The stated nondegeneracy excludes a=b=c, where the second direction would be zero.
  3. Both are nonzero and have zero dot product; hence the angle is π/2.

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