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NCERT Class 12 · Ch 11 Exercise 11.2 · Q9

NCERT Class 12 Maths · Chapter 11 Three Dimensional Geometry · Exercise 11.2 Question 9

Find acute angles between (i) (x−2)/2=(y−1)/5=(z+3)/(−3) and (x+2)/(−1)=(y−4)/8=(z−5)/4; (ii) x/2=y/2=z and (x−5)/4=(y−2)=(z−3)/8.

💡 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: (i) cos⁻¹(13*sqrt(38)/171); (ii) cos⁻¹(2/3)
  1. The acute line angle satisfies cos θ=|d·e|/(|d||e|).
  2. (i) d=(2,5,-3), e=(-1,8,4), dot product 26, norms sqrt(38) and 9. Thus θ=cos⁻¹(13*sqrt(38)/171).
  3. (ii) d=(2,2,1), e=(4,1,8), dot product 18, norms 3 and 9. Thus θ=cos⁻¹(2/3).

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