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Class 12 Three Dimensional Geometry · Long Answer3 Marksorthogonal_directions

Class 12 Maths · Three Dimensional Geometry · Long Answer #45

Find vector and Cartesian equations of a line through (1,2,-4) whose direction is orthogonal to (3,-16,7) and (3,8,-5)

✅ Step-by-Step Solution

Final Answer: r=(1,2,-4)+t(2,3,6), t∈ℝ; (x−(1))/(2)=(y−(2))/(3)=(z−(-4))/(6).
  1. A common orthogonal direction is d×e.
  2. d=(3,-16,7), e=(3,8,-5), d×e=(24,36,72).
  3. Take the proportional nonzero direction n=(2,3,6); n·d=n·e=0.
  4. r=(1,2,-4)+t(2,3,6), t∈ℝ; (x−(1))/(2)=(y−(2))/(3)=(z−(-4))/(6).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] A common orthogonal direction is d×e.
  • [1 mark] d=(3,-16,7), e=(3,8,-5), d×e=(24,36,72).
  • [1 mark] Take the proportional nonzero direction n=(2,3,6); n·d=n·e=0.
  • [0.5 marks] r=(1,2,-4)+t(2,3,6), t∈ℝ; (x−(1))/(2)=(y−(2))/(3)=(z−(-4))/(6).

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