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7.Angle Between Lines
Class 12 • Mathematics • NCERT
Angle Between Two Lines (3D Geometry)
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This lesson explains how to find the angle between two lines in three-dimensional space using direction ratios and direction cosines, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand angle between two lines in 3D.
- Use direction ratios to find the angle.
- Apply the standard NCERT formula.
- Solve board-level problems.
1. Angle Between Two Lines
The angle between two lines in space is defined as the angle between their direction vectors.
Let θ be the angle between the two given lines.
[ Image Placeholder: Two Lines Making an Angle in 3D ]
2. Direction Ratios of Lines
The direction ratios of a line represent the components of its direction vector.
If line L₁ has DRs (a₁, b₁, c₁)
and line L₂ has DRs (a₂, b₂, c₂)
and line L₂ has DRs (a₂, b₂, c₂)
[ Image Placeholder: Direction Ratios of Two Lines ]
3. Formula for Angle Between Two Lines
The cosine of the angle between two lines is given by:
cosθ = (a₁a₂ + b₁b₂ + c₁c₂) / √[(a₁² + b₁² + c₁²)(a₂² + b₂² + c₂²)]
[ Image Placeholder: Formula Application Diagram ]
4. Special Cases
Certain values of θ give special relationships between lines.
• If cosθ = 0 → Lines are perpendicular
• If cosθ = 1 → Lines are parallel
• If cosθ = 1 → Lines are parallel
[ Image Placeholder: Parallel and Perpendicular Lines ]
5. NCERT Solved Example
Find the angle between the lines with DRs (1, 2, 2) and (2, −1, 2).
cosθ = (1×2 + 2×(−1) + 2×2) / √[(1²+2²+2²)(2²+(−1)²+2²)]
= (2 − 2 + 4) / √[(9)(9)]
= 4/9
[ Image Placeholder: Worked Example – Angle Between Lines ]
6. Important NCERT Notes
• Angle is always acute unless stated otherwise
• Use direction ratios carefully
• Formula is symmetric in both lines
• Very common board-exam question
• Use direction ratios carefully
• Formula is symmetric in both lines
• Very common board-exam question
Practice Questions (NCERT)
- Define angle between two lines in 3D.
- Write the formula for angle between two lines.
- What are direction ratios?
- When are two lines perpendicular?
- When are two lines parallel?
- Find angle if cosθ = 0.
- Does order of lines matter in formula?
- Find angle between lines with DRs (1,1,1) and (1,−1,1).
- Is this topic important for board exams?
- Is this formula part of NCERT syllabus?
✅ Show Answer Key
- Angle between direction vectors
- cosθ formula using DRs
- Components of direction vector
- cosθ = 0
- cosθ = 1
- 90°
- No
- cosθ = 1/3
- Yes
- Yes
© Aviate Learning – Angle Between Lines (NCERT Class 12)