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Class 12 Three Dimensional Geometry · Very Long Answer5 Marksparallel_lines

Class 12 Maths · Three Dimensional Geometry · Very Long Answer #53

Find vector and Cartesian equations through P(1,2,−4) parallel to the line through A(3,3,−5),B(1,0,−11). Also find the distance between these lines.

✅ Step-by-Step Solution

Final Answer: r=(1,2,−4)+t(2,3,6); (x−1)/2=(y−2)/3=(z+4)/6; distance √293/7.
  1. B−A=(−2,−3,−6), so use direction d=(2,3,6).
  2. The required vector line is r=(1,2,−4)+td.
  3. Eliminating t gives (x−1)/2=(y−2)/3=(z+4)/6.
  4. A−P=(2,1,−1); (A−P)×d=(9,−14,4), squared norm 293, while |d|=7.
  5. Distance is √293/7.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] B−A=(−2,−3,−6), so use direction d=(2,3,6).
  • [1 mark] The required vector line is r=(1,2,−4)+td.
  • [1 mark] Eliminating t gives (x−1)/2=(y−2)/3=(z+4)/6.
  • [1 mark] A−P=(2,1,−1); (A−P)×d=(9,−14,4), squared norm 293, while |d|=7.
  • [1 mark] Distance is √293/7.

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