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Class 12 Three Dimensional Geometry · Long Answer3 Marksorthogonal_directions

Class 12 Maths · Three Dimensional Geometry · Long Answer #85

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

✅ Step-by-Step Solution

Final Answer: r=(1,-1,1)+t(-10,4,7), t∈ℝ; (x−(1))/(-10)=(y−(-1))/(4)=(z−(1))/(7).
  1. A common orthogonal direction is d×e.
  2. d=(-3,-4,-2), e=(1,-1,2), d×e=(-10,4,7).
  3. Take the proportional nonzero direction n=(-10,4,7); n·d=n·e=0.
  4. r=(1,-1,1)+t(-10,4,7), t∈ℝ; (x−(1))/(-10)=(y−(-1))/(4)=(z−(1))/(7).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] A common orthogonal direction is d×e.
  • [1 mark] d=(-3,-4,-2), e=(1,-1,2), d×e=(-10,4,7).
  • [1 mark] Take the proportional nonzero direction n=(-10,4,7); n·d=n·e=0.
  • [0.5 marks] r=(1,-1,1)+t(-10,4,7), t∈ℝ; (x−(1))/(-10)=(y−(-1))/(4)=(z−(1))/(7).

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