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Class 12 Three Dimensional Geometry · Very Long Answer5 Marksdirection_cosines

Class 12 Maths · Three Dimensional Geometry · Very Long Answer #61

Let a=(l₁,m₁,n₁), b=(l₂,m₂,n₂), c=(l₃,m₃,n₃) be mutually perpendicular unit direction vectors. Prove that the line with direction a+b+c makes equal acute angles with the three given lines.

✅ Step-by-Step Solution

Final Answer: Each acute angle is cos⁻¹(1/√3).
  1. Mutual perpendicularity gives a·b=b·c=c·a=0, and each squared norm is 1.
  2. Set d=a+b+c. Then |d|²=3, so d is nonzero and |d|=√3.
  3. Compute d·a=d·b=d·c=1.
  4. Thus each absolute normalized dot product is 1/√3.
  5. All three acute angles equal cos⁻¹(1/√3).

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Mutual perpendicularity gives a·b=b·c=c·a=0, and each squared norm is 1.
  • [1 mark] Set d=a+b+c. Then |d|²=3, so d is nonzero and |d|=√3.
  • [1 mark] Compute d·a=d·b=d·c=1.
  • [1 mark] Thus each absolute normalized dot product is 1/√3.
  • [1 mark] All three acute angles equal cos⁻¹(1/√3).

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