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Class 12 Three Dimensional Geometry · Long Answer3 Marksorthogonal_directions

Class 12 Maths · Three Dimensional Geometry · Long Answer #47

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

✅ Step-by-Step Solution

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

📋 CBSE Stepwise Marking Scheme

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

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