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4.Direction Cosines & Ratios
Class 12 • Mathematics • NCERT
Direction Cosines & Direction Ratios
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This lesson explains direction cosines and direction ratios of a line in space, which describe the orientation of a vector or a line with respect to coordinate axes, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand direction angles and direction cosines.
- Learn the relation between direction cosines.
- Understand direction ratios and their use.
- Solve NCERT board-level problems.
1. Direction Angles
The angles made by a directed line with the positive directions of the x-axis, y-axis, and z-axis are called direction angles.
Let α, β, γ be the angles with x, y, z axes respectively.
[ Image Placeholder: Line Making Angles with Coordinate Axes ]
2. Direction Cosines
The cosines of the direction angles are called direction cosines.
l = cosα, m = cosβ, n = cosγ
[ Image Placeholder: Direction Cosines l, m, n ]
3. Relation Between Direction Cosines
The direction cosines of a line satisfy an important relation.
l² + m² + n² = 1
[ Image Placeholder: Geometric Interpretation of l² + m² + n² = 1 ]
4. Direction Ratios
Any three numbers proportional to the direction cosines of a line are called direction ratios.
If direction cosines are l, m, n, then direction ratios are a, b, c where:
l = a/√(a² + b² + c²)
l = a/√(a² + b² + c²)
[ Image Placeholder: Direction Ratios of a Line ]
5. Direction Cosines from Direction Ratios
Direction cosines can be obtained from given direction ratios.
If direction ratios are a, b, c, then:
l = a / √(a² + b² + c²),
m = b / √(a² + b² + c²),
n = c / √(a² + b² + c²)
l = a / √(a² + b² + c²),
m = b / √(a² + b² + c²),
n = c / √(a² + b² + c²)
[ Image Placeholder: Converting DRs to DCs ]
6. Important NCERT Notes
• Direction cosines are unique for a given line
• Direction ratios are not unique
• Relation l² + m² + n² = 1 is compulsory
• Used extensively in 3D geometry
• Direction ratios are not unique
• Relation l² + m² + n² = 1 is compulsory
• Used extensively in 3D geometry
Practice Questions (NCERT)
- Define direction angles.
- What are direction cosines?
- State the relation between direction cosines.
- Are direction ratios unique?
- Find direction cosines of a line with DRs 2, −2, 1.
- What is the relation between DRs and DCs?
- Can direction cosines be negative?
- How many direction cosines does a line have?
- What do direction ratios represent?
- Is this topic important for board exams?
✅ Show Answer Key
- Angles with coordinate axes
- Cosines of direction angles
- l² + m² + n² = 1
- No
- 2/3, −2/3, 1/3
- DCs are normalized DRs
- Yes
- Three
- Orientation of line
- Yes
© Aviate Learning – Direction Cosines & Ratios (NCERT Class 12)