3D Geometry Class 11 Math Quick Recall / Short Notes
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[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)
- Coordinate Planes: XY plane (z=0), YZ plane (x=0), ZX plane (y=0).
- Octants: The space is divided into 8 octants by the three planes.
- Distance Formula (P₁P₂): √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].
- Section Formula (Internal): [(mx₂ + nx₁) / (m+n), (my₂ + ny₁) / (m+n), (mz₂ + nz₁) / (m+n)].
- Centroid of a Triangle: [(x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3]. 📥 Download 1-Page Short Notes PDF (Zero-Friction)
Introduction
Three Dimensional Geometry extends the Cartesian coordinate system into space, providing the foundation for engineering, flight navigation, and Vector Algebra. Master the Eight Octants, the 3D distance formula, and section formulas to excel in spatial modeling and advanced physics. This Class 11 Math Chapter 12 summary provides all essential formulas for JEE and Board exam success. The transition from 2D to 3D geometry is like going from a flat map to the real world.
1. Coordinate Planes and Axes in 3D
In three dimensions, we use three mutually perpendicular lines passing through the origin: the X, Y, and Z axes.
- Coordinate Planes:
- XY Plane: Contains x and y axes. Equation: z = 0.
- YZ Plane: Contains y and z axes. Equation: x = 0.
- ZX Plane: Contains z and x axes. Equation: y = 0.
2. Coordinates of a Point and Octants
A point P in space is represented by (x, y, z). These axes divide the space into 8 octants.
- Sign Convention for Octants:
- I: (+, +, +)
- II: (-, +, +)
- III: (-, -, +)
- IV: (+, -, +)
- V: (+, +, -)
- VI: (-, +, -)
- VII: (-, -, -)
- VIII: (+, -, -)
3. Distance Formula in 3D
The distance between two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) is given by: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] This formula is an extension of the Pythagorean theorem into three dimensions.
4. Section Formula
If a point R divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in the ratio m : n:
- Internal Division: R = [(mx₂ + nx₁) / (m+n), (my₂ + ny₁) / (m+n), (mz₂ + nz₁) / (m+n)]
- External Division: R = [(mx₂ - nx₁) / (m-n), (my₂ - ny₁) / (m-n), (mz₂ - nz₁) / (m-n)]
- Midpoint: M = [(x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2]
5. Centroid of a Triangle
The centroid of a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), and (x₃, y₃, z₃) is: G = [(x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3, (z₁ + z₃ + z₃) / 3]
Comprehensive Exam Strategy (Q&A)
Q1: Name the octant in which the point (-3, 1, -2) lies. Answer:
- x is negative, y is positive, z is negative.
- Sign pattern: (-, +, -).
- This corresponds to Octant VI.
Q2: Find the distance between P(1, -3, 4) and Q(-4, 1, 2). Answer:
- d = √[(-4 - 1)² + (1 - (-3))² + (2 - 4)²]
- d = √[(-5)² + (4)² + (-2)²]
- d = √[25 + 16 + 4] = √45
- d = 3√5 units.
Q3: Find the coordinates of the point which divides the line joining (1, -2, 3) and (3, 4, -5) internally in the ratio 2 : 3. Answer:
- x = [2(3) + 3(1)] / 5 = 9/5
- y = [2(4) + 3(-2)] / 5 = 2/5
- z = [2(-5) + 3(3)] / 5 = -1/5
- Point = (9/5, 2/5, -1/5).
Related Revision Notes
- Chapter 11: Conic Sections
- Chapter 13: Limits and Derivatives
- [External Reference: NCERT Class 11 Math Chapter 12 (Authoritative Source)]
Conclusion
3D Geometry is the gateway to understanding spatial relationships. By mastering the octant signs and adapting the distance and section formulas from 2D to 3D, you build the mental framework required for advanced physics and engineering. Always visualize the point relative to the coordinate planes, and remember that x, y, or z being zero tells you exactly which plane you're on! Keep your spatial orientation clear.