Introduction to Three Dimensional Geometry for JEE Main

The chapter Three Dimensional Geometry (3D Geometry) is a fundamental topic in Class 11 Maths, forming the foundation for vectors, calculus, and spatial reasoning problems in JEE Main.

3D Geometry deals with points, lines, and planes in three-dimensional space using coordinates (x, y, z). Understanding distance, section, and direction formulas is essential for solving numerical and conceptual problems efficiently.

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Coordinate System in 3D

1. Coordinates of a Point:

• A point P in 3D space is represented as P(x, y, z)

• x, y, z = distances from yz-plane, xz-plane, xy-plane respectively

2. Axes and Planes:

• x-axis, y-axis, z-axis: mutually perpendicular

• Coordinate planes: xy-plane (z = 0), yz-plane (x = 0), xz-plane (y = 0)

3. Origin:

• Point O(0, 0, 0)

Distance Formula in 3D

• Distance between two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂):
  d = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²]

Example: Distance between A(1, 2, 3) and B(4, 6, 8) → √[(4–1)² + (6–2)² + (8–3)²] = √(9 + 16 + 25) = √50

Section Formula in 3D

• Coordinates of point P dividing line segment AB internally in ratio m:n:
  P = [(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)]

• Externally dividing point: P = [(mx₂ – nx₁)/(m – n), …]

Example: Point dividing A(1, 2, 3) and B(4, 6, 8) in ratio 2:1 → P = [(2×4 + 1×1)/3, (2×6 + 1×2)/3, (2×8 + 1×3)/3] = (3, 14/3, 19/3)

Direction Cosines and Ratios

1. Direction Cosines:

• Cosines of angles made by line with x, y, z axes

• If line has direction ratios l, m, n → l² + m² + n² = 1

2. Direction Ratios:

• Proportional to direction cosines

• Used to define line in space

Distance from a Point to Coordinate Planes

• Distance from point P(x₀, y₀, z₀) to:

  • xy-plane: |z₀|

  • yz-plane: |x₀|

  • xz-plane: |y₀|

• Distance from point to origin: √(x₀² + y₀² + z₀²)

Angle Between Two Lines in 3D

• If direction ratios of lines are (l₁, m₁, n₁) and (l₂, m₂, n₂):
  cos θ = (l₁l₂ + m₁m₂ + n₁n₂) / [√(l₁² + m₁² + n₁²) × √(l₂² + m₂² + n₂²)]

• Special cases:

  • Lines perpendicular → cos θ = 0

  • Lines parallel → cos θ = ±1

Distance Between Two Points on Space Diagonal

• Applications in cube, rectangular prism, and spatial problems

• Use distance formula in 3D for solving JEE Main questions

Applications of 3D Geometry in JEE Main

• Solve distance and section problems in space

• Find angle between lines, planes, and coordinate axes

• Basis for vector problems, planes, and conic sections in higher classes

• Frequently tested in MCQs, numerical, and integer-type questions

Example: Find distance between points P(1, 2, 3) and Q(4, 0, –1) → √[(4–1)² + (0–2)² + (–1–3)²] = √(9 + 4 +16) = √29

Important JEE Main Preparation Tips for 3D Geometry

• Memorize distance, section, and direction formulas

• Practice finding coordinates, ratios, and distance from planes

• Solve angle between lines and space problems

• Visualize 3D points and lines for faster solutions

• Solve previous year JEE Main MCQs and numerical problems

Why Study 3D Geometry from Studentbro.in?

Studentbro.in provides:

• Step-by-step explanations of 3D coordinates, distance, and section formulas

• Solved examples for direction ratios, angles, and planes

• Diagrams and charts for quick revision

• MCQs and PYQs aligned with JEE Main syllabus

Conclusion

The Introduction to Three Dimensional Geometry chapter is fundamental and scoring in Class 11 Maths for JEE Main. Mastery of coordinates, distance, section, direction cosines, and angles helps students solve conceptual and numerical problems efficiently.

Studentbro.in provides structured, easy-to-understand, and exam-focused content to master 3D Geometry effectively and boost JEE Main scores.