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5.Equation of Line in 3D
Class 12 • Mathematics • NCERT
Equation of a Line in 3D
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This lesson explains different forms of the equation of a line in three-dimensional space, including vector, parametric, and symmetric forms, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand the equation of a line in vector form.
- Learn parametric and symmetric forms of a line.
- Use direction ratios and direction cosines.
- Solve NCERT board-level problems.
1. Equation of a Line in Vector Form
The vector equation of a line passing through a fixed point with position vector →a and having direction vector →b is:
→r = →a + λ→b
[ Image Placeholder: Line Through a Point in Vector Form ]
2. Direction Ratios of a Line
The direction ratios of a line are any three numbers proportional to the components of its direction vector.
If →b = ai + bj + ck, then a, b, c are direction ratios.
[ Image Placeholder: Direction Ratios of a Line ]
3. Parametric Form of Equation of Line
The parametric equations of a line passing through point (x₁, y₁, z₁) with direction ratios (a, b, c) are:
x = x₁ + aλ
y = y₁ + bλ
z = z₁ + cλ
y = y₁ + bλ
z = z₁ + cλ
[ Image Placeholder: Parametric Representation of a Line ]
4. Symmetric Form of Equation of Line
The symmetric form of the equation of a line is obtained by eliminating the parameter λ.
(x − x₁)/a = (y − y₁)/b = (z − z₁)/c
[ Image Placeholder: Symmetric Form of Line ]
5. Line Passing Through Two Points
The equation of a line passing through points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is:
(x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁) = (z − z₁)/(z₂ − z₁)
[ Image Placeholder: Line Through Two Points ]
6. Important NCERT Notes
• Vector form is the most fundamental form
• Direction ratios need not be unique
• Symmetric form is useful in problem-solving
• Very important chapter for board exams
• Direction ratios need not be unique
• Symmetric form is useful in problem-solving
• Very important chapter for board exams
Practice Questions (NCERT)
- Write the vector equation of a line.
- What are direction ratios?
- Write parametric equations of a line.
- Write symmetric form of a line.
- Find equation of line through (1,2,3) with DRs (2,−1,1).
- Write equation of line through two points.
- Which form is most fundamental?
- Are direction ratios unique?
- Can direction ratios be zero?
- Is this topic important for board exams?
✅ Show Answer Key
- →r = →a + λ→b
- Numbers proportional to direction vector
- x = x₁ + aλ, etc.
- (x − x₁)/a = (y − y₁)/b = (z − z₁)/c
- (x−1)/2 = (y−2)/−1 = (z−3)/1
- Symmetric form
- Vector form
- No
- Yes
- Yes
© Aviate Learning – Equation of Line in 3D (NCERT Class 12)