Class 12 Maths Chapter 11 Three-Dimensional Geometry Notes FREE PDF Download
FAQs on Three-Dimensional Geometry Class 12 Notes: CBSE Maths Chapter 11
1. What are the key concepts covered in the Three-Dimensional Geometry Class 12 Revision Notes as per the CBSE 2025–26 syllabus?
The Three-Dimensional Geometry Class 12 Revision Notes summarise core concepts such as direction cosines and ratios, equations of lines and planes, angle calculation between lines and planes, shortest distance formulas, and applications of distance and midpoint formulas in 3D space. These notes follow the official CBSE 2025–26 curriculum, enabling quick revision of all essential topics required for board exam preparation.
2. How should students use revision notes to prepare effectively for Chapter 11 of Class 12 Maths?
To revise Chapter 11: Three-Dimensional Geometry efficiently, begin by reviewing the summary and all key formulas in the notes. Next, practice sample problems, particularly those involving equations of lines and planes. Establish connections between concepts, such as how direction ratios relate to direction cosines. Regular recap using revision notes and solving previous year questions can consolidate understanding before exams.
3. Which formulas must be memorised from Class 12 Three-Dimensional Geometry for quick revision?
- Distance Formula: \(AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\)
- Midpoint Formula: \(M = ((x_1 + x_2)/2, (y_1 + y_2)/2, (z_1 + z_2)/2)\)
- Equation of a Plane: \(a(x-x_1)+b(y-y_1)+c(z-z_1)=0\)
- Distance of Point from Plane: \(|ax_1 + by_1 + cz_1 + d|/\sqrt{a^2 + b^2 + c^2}\)
- Equation of a line: Vector form: \(\vec{r} = \vec{a} + \lambda\vec{b}\); Cartesian form: \((x-x_1)/a = (y-y_1)/b = (z-z_1)/c\)
4. What is the best order to revise the topics of Three-Dimensional Geometry for exams?
Start with coordinates in 3D space, progress to distance and midpoint formulas, then cover direction cosines and direction ratios. After mastering line and plane equations, move to angle calculation between lines and planes, and finally cover shortest distance and intersection. This logical sequence helps in building foundational understanding before tackling application-based problems.
5. How are the direction cosines and direction ratios of a line related, and why is this relationship important for quick revision?
Direction ratios (DRs: a, b, c) are proportional numbers indicating the direction of a line, while direction cosines (DCs: l, m, n) are the cosines of angles the line makes with the coordinate axes. Their relation is:
- \(l = a/\sqrt{a^2+b^2+c^2}, m = b/\sqrt{a^2+b^2+c^2}, n = c/\sqrt{a^2+b^2+c^2}\)
- Also, \(l^2 + m^2 + n^2 = 1\)
6. What are the most common mistakes students make when revising Three-Dimensional Geometry, and how can they be avoided?
Common pitfalls include confusing direction ratios with coordinates, forgetting to normalise direction cosines, and mixing up equations for lines and planes. To avoid these, always refer to the key formulas and revise the stepwise derivation for each type of equation from the notes. Practice applying formulas to different types of problems and double-check units and vector/cosine requirements.
7. How can students visualise 3D geometry problems for better retention during revision?
Students should draw diagrams of lines, planes, and points in three-dimensional space while studying. Using graph paper or digital tools for sketching, and labelling axes, intersections, and vectors, supports memory and aids in comprehending spatial relationships – a crucial aspect for questions involving projections, intersections, and shortest distances.
8. What types of questions can appear based on the CBSE's focus for Three-Dimensional Geometry revision?
- Short answer questions on writing equations of lines or planes
- Problems that require calculating distances (point-to-point, point-to-plane, line-to-line)
- Proving lines are parallel, perpendicular, or skew
- Finding direction cosines or ratios given geometric conditions
- Application-type problems connecting multiple formulas, such as shortest distance between skew lines
9. Why is the concept of skew lines essential in Class 12 Three-Dimensional Geometry, and how does it connect with other topics?
Skew lines are lines that do not intersect and are not parallel (i.e., they do not lie in the same plane). Their study is essential because their shortest distance calculation involves vector cross products and properties of both lines and planes. This concept interlinks all major areas—line equations, vector products, and spatial relations—and is common in higher-order board exam questions.
10. What are efficient strategies for last-minute revision of Three-Dimensional Geometry before the Class 12 board exam?
- Revisit all key formulas and the summary at the start of the revision notes.
- Practice 1–2 representative problems of each type: line/plane equations, distances, and angle calculations.
- Review example errors and FUQs from the notes to clarify conceptual gaps.
- Use visualisation for confusing spatial problems.
- Target tricky sections like intersection forms or angle between skew lines, as these often appear in board exams.

















