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NCERT Class 12 · Ch 11 Exercise 11.2 · Q8

NCERT Class 12 Maths · Chapter 11 Three Dimensional Geometry · Exercise 11.2 Question 8

Find acute angles between (i) r=(2,−5,1)+t(3,2,6) and r=(7,0,−6)+s(1,2,2); (ii) r=(3,1,−2)+t(1,−1,−2) and r=(2,−1,−56)+s(3,−5,−4).

💡 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(19/21); (ii) cos⁻¹(8*sqrt(3)/15)
  1. The acute line angle satisfies cos θ=|d·e|/(|d||e|).
  2. (i) d=(3,2,6), e=(1,2,2), dot product 19, norms 7 and 3. Thus θ=cos⁻¹(19/21).
  3. (ii) d=(1,-1,-2), e=(3,-5,-4), dot product 16, norms sqrt(6) and 5*sqrt(2). Thus θ=cos⁻¹(8*sqrt(3)/15).

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