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6.Distance Between Points
Class 12 • Mathematics • NCERT
Distance Between Points (3D Geometry)
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This lesson explains how to find the distance between two points in three-dimensional space using the distance formula, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand distance formula in 3D.
- Apply distance formula between two points.
- Relate 3D distance with 2D distance.
- Solve NCERT board-level problems.
1. Distance Between Two Points in 3D
The distance between two points in three-dimensional space is an extension of the distance formula in two dimensions.
Let A(x₁, y₁, z₁) and B(x₂, y₂, z₂) be two points.
[ Image Placeholder: Two Points in 3D Coordinate System ]
2. Distance Formula
The distance between points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is given by:
AB = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
[ Image Placeholder: Distance Formula in 3D ]
3. Distance from Origin
The distance of a point from the origin is a special case of the distance formula.
If A(x, y, z), then OA = √(x² + y² + z²)
[ Image Placeholder: Distance of a Point from Origin ]
4. Relation with 2D Distance Formula
The 3D distance formula reduces to the 2D distance formula when z-coordinates are zero.
Distance in 2D: √[(x₂ − x₁)² + (y₂ − y₁)²]
[ Image Placeholder: Comparison of 2D and 3D Distance ]
5. NCERT Solved Example
Find the distance between A(1, 2, 3) and B(4, 6, 3).
AB = √[(4−1)² + (6−2)² + (3−3)²]
= √(9 + 16 + 0)
= 5 units
[ Image Placeholder: Worked Distance Example ]
6. Important NCERT Notes
• Distance formula is symmetric
• Distance is always non-negative
• Square root must be simplified carefully
• Very common board-exam questions
• Distance is always non-negative
• Square root must be simplified carefully
• Very common board-exam questions
Practice Questions (NCERT)
- Write the distance formula in 3D.
- Find the distance between (1,0,2) and (4,0,6).
- Find distance of point (3,4,12) from origin.
- Is distance ever negative?
- What happens to formula if z = 0?
- Find distance between (2,3,1) and (2,3,1).
- Does order of points affect distance?
- Find distance between (0,0,0) and (a,b,c).
- Is this formula part of NCERT Class 12?
- Is this chapter important for board exams?
✅ Show Answer Key
- √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]
- 5
- 13
- No
- Becomes 2D formula
- 0
- No
- √(a²+b²+c²)
- Yes
- Yes
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