Three Dimensional Geometry studies the position and relationship of points, lines, and planes in three-dimensional space using the x, y, and z coordinate system.
Direction ratios are numbers proportional to the direction of a line, while direction cosines are the normalized values of those ratios and satisfy (l^2+m^2+n^2=1).
Vectors simplify the representation of points, lines, distances, and angles in space and make solving three-dimensional problems more efficient.
The shortest distance is the minimum perpendicular distance between two non-intersecting and non-parallel lines in three-dimensional space.
Three Dimensional Geometry is a high-weightage chapter that frequently features conceptual and numerical questions involving lines, vectors, distances, and angles, making it one of the most scoring topics in Mathematics.
Join ALLEN!
(Session 2026 - 27)
Choose class
Choose your goal
Preferred Mode
Choose State
Class 12 Maths Chapter 11: Three Dimensional Geometry
Three Dimensional Geometry (3D Geometry) extends coordinate geometry from two dimensions to three dimensions by introducing the x, y, and z coordinate system. This chapter focuses on representing points, direction ratios, direction cosines, equations of lines, shortest distance between lines, and angles between lines. It is one of the most important chapters in CBSE Board, JEE Main, and JEE Advanced, where questions are often formula-based as well as application-oriented.
At ALLEN, these revision notes simplify Three Dimensional Geometry through concise explanations, essential formulas, graphical interpretation, solved examples, and exam-oriented tips. Class 12 Maths Chapter 11 revision notes help students strengthen conceptual understanding and improve problem-solving speed for board and competitive examinations.
Angle Between Two Lines: cosθ=a12+b12+c12a22+b22+c22a1a2+b1b2+c1c2
Important Properties / Shortcut Tips
Convert direction ratios into direction cosines whenever required.
Remember that direction cosines always satisfy (l^2+m^2+n^2=1).
Use vector notation to simplify line equations.
Check whether lines are parallel or perpendicular before lengthy calculations.
Memorize standard formulas for angles and distances.
5.0High Weightage Topics
Topic
JEE Importance
Difficulty Level
Direction Ratios & Direction Cosines
High
Easy–Moderate
Equation of a Line
Very High
Moderate
Angle Between Two Lines
Very High
Moderate
Shortest Distance Between Two Lines
Very High
Moderate–High
Coplanarity of Lines
High
Moderate
Distance Between Point & Line
High
Moderate
Mixed 3D Geometry Problems
Very High
High
6.0Formula Handbook
Available soon
7.0Common Mistakes & JEE Tips
Common Mistake
Correct Approach
Confusing direction ratios with direction cosines
Direction cosines are normalized values of direction ratios.
Forgetting to normalize vectors
Convert DRs into DCs whenever required.
Using incorrect line equations
Identify the correct point and direction vector before forming the equation.
Sign errors in distance calculations
Apply the distance formula carefully.
Incorrect angle calculation
Use the dot product formula accurately.
Missing vector notation
Write equations systematically in vector or Cartesian form.
Algebraic simplification errors
Simplify expressions before substituting values.
8.0ALLEN Faculty Tips
Memorize all standard 3D Geometry formulas.
Practice converting direction ratios into direction cosines.
Revise vector algebra alongside this chapter.
Solve angle and shortest-distance problems regularly.
Draw rough diagrams to visualize spatial relationships.
Practice Previous Years' Questions (PYQs) for speed and accuracy.
9.0PYQ Trend Analysis
Three Dimensional Geometry is one of the highest-scoring chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Most questions are formula-based and involve direction ratios, equations of lines, angles, and shortest distances.
Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.
Topic
Questions Asked (Last 5 Years)
Average Difficulty
Direction Ratios & Cosines
1
Easy
Equation of Line
1–2
Moderate
Angle Between Two Lines
1
Moderate
Shortest Distance
1–2
Moderate–High
Mixed 3D Problems
1
Moderate–High
10.0Most Common Question Types
Finding direction ratios and direction cosines.
Writing equations of lines in different forms.
Finding the angle between two lines.
Calculating the shortest distance between skew lines.
Determining the distance between a point and a line.
Solving coplanarity-based problems.
Applying vector methods in three-dimensional geometry.
Solving mixed conceptual and numerical questions.
11.0Smart Revision Strategy
Revise vector algebra before studying this chapter.
Memorize all standard 3D Geometry formulas.
Practice line equation problems.
Solve direction ratio and direction cosine questions.