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Class 12 Three Dimensional Geometry · Very Long Answer5 Marksprojection_and_reflection

Class 12 Maths · Three Dimensional Geometry · Very Long Answer #89

Find the foot and distance from P=(2,1,2) to r=(4,2,3)+t(1,-1,-1), t∈ℝ. Also find its reflection in this line.
Perpendicular projection and reflection
Perpendicular projection and reflection

✅ Step-by-Step Solution

Final Answer: Foot H=(4,2,3); distance=sqrt(6). Reflection=(6,3,4).
  1. Write H=A+td and impose (P−H)·d=0.
  2. A=(4,2,3), d=(1,-1,-1), P−A=(-2,-1,-1); t=((P−A)·d)/(d·d)=0.
  3. H=A+td=(4,2,3). Check (P−H)·d=0.
  4. |P−H|=√(6)=sqrt(6). Reflection I=2H−P=(6,3,4).
  5. Foot H=(4,2,3); distance=sqrt(6). Reflection=(6,3,4).

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Write H=A+td and impose (P−H)·d=0.
  • [1 mark] A=(4,2,3), d=(1,-1,-1), P−A=(-2,-1,-1); t=((P−A)·d)/(d·d)=0.
  • [1 mark] H=A+td=(4,2,3). Check (P−H)·d=0.
  • [1 mark] |P−H|=√(6)=sqrt(6). Reflection I=2H−P=(6,3,4).
  • [1 mark] Foot H=(4,2,3); distance=sqrt(6). Reflection=(6,3,4).

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