Angle Between Lines
Mediummathematics
If θ is angle between two lines with direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂), then cos θ equals
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Solution
Incorrect! Answer:
l₁l₂ + m₁m₂ + n₁n₂
- Vector Representation: If lines have direction cosines (l1,m1,n1) and (l2,m2,n2), then unit vectors along the lines are u1=l1i^+m1j^+n1k^ and u2=l2i^+m2j^+n2k^.
- Dot Product: cosθ=∣u1∣∣u2∣u1⋅u2.
- Simplification: Since they are unit vectors, ∣u1∣=∣u2∣=1. Thus, u1⋅u2=l1l2+m1m2+n1n2.
- Result: cosθ=l1l2+m1m2+n1n2.
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About This Question
- Subject
- mathematics
- Chapter
- three dimensional geometry
- Topic
- angle between lines
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
l₁l₂ + m₁m₂ + n₁n₂
- Vector Representation: If lines have direction cosines (l1,m1,n1) and (l2,m2,n2), then unit vectors along the lines are u1=l1i^+m1j^+n1k^ and u2=l2i^+m2j^+n2k^.
- Dot Product: cosθ=∣u1∣∣u2∣u1⋅u2.
- Simplification: Since they are unit vectors, ∣u1∣=∣u2∣=1. Thus, u1⋅u2=l1l2+m1m2+n1n2.
- Result: cosθ=l1l2+m1m2+n1n2.
This medium difficulty mathematics question is from the chapter three dimensional geometry, covering the topic of angle between lines. It appeared in the 2025 exam.
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