NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry
Three Dimensional Geometry is one of those chapters where students either fall in love with the subject or struggle to picture what is actually going on, mostly because everything has shifted from a flat page to a space with depth. Myclass24 has prepared these NCERT Solutions for Class 12 Maths Chapter 11 to help you see the geometry behind the algebra, not just memorise formulas for direction cosines or the angle between two lines. Every solution is worked out step by step the way a teacher would explain it on a board, so that you understand why a particular formula applies to a particular question, not only how to plug in numbers.
Summary of Chapter 11 Three-dimensional Geometry
Introduction, Direction cosines, direction ratios, vector, and Cartesian equation of a line in space. The angle between two lines, Skew lines, is the shortest distance between skew lines and parallel lines. Plane: Introduction, Various equations of the plane, Family of planes Coplanar lines, angle between planes, Plane, and line– Angle, Distance of a point from a plane and a line. Image of a point concerning a plane
Find the PDF of NCERT Solutions for Class 12 Maths Chapter 11
Students preparing for boards or competitive exams often want a clean, offline copy of these solutions to revise on the night before a test or while travelling. The downloadable NCERT Solutions for Class 12 PDF from Myclass24 contains every question from the chapter solved in the NCERT-prescribed method, along with extra working steps that the textbook itself skips. It is formatted for quick scanning, with each exercise kept separate so you are not hunting through pages to find Exercise 11.2 when you only need that one.
Class | 12 |
Subject | Mathematics |
Chapter Number | 11 |
Chapter Name | Three Dimensional Geometry |
Board | CBSE / NCERT |
Total Exercises | 5 (including Miscellaneous Exercise) |
Key Topics | Direction cosines and ratios, equation of a line in space, angle between two lines, shortest distance between two lines, equation of a plane, angle between two planes, distance of a point from a plane |
Weightage | Usually combined with Vectors for around 6 to 10 marks in the board exam |
Difficulty Level | Moderate, becomes easy once direction cosines are clear |
Prerequisite Chapter | Chapter 10 (Vector Algebra) |
Understanding Chapter 11: What Actually Trips Students Up
If you ask most Class 12 students where they lose marks in this chapter, the honest answer is usually the same place: confusing direction cosines with direction ratios, and forgetting that direction cosines must satisfy l squared plus m squared plus n squared equals 1, while direction ratios can be any proportional set of numbers. Myclass24's solutions deliberately slow down at this exact point in every question, showing the conversion from ratios to cosines explicitly rather than skipping the normalisation step the way some guides do. Once this distinction is internalised, half the chapter's difficulty disappears on its own.
The second recurring trouble spot is the shortest distance between two skew lines. Students often get the vector form correct but make sign errors while computing the cross product, or they forget that if the lines intersect, the shortest distance comes out to zero and that itself is the answer, not an error in their working. Our solutions walk through the determinant expansion for the cross product line by line so you can see exactly where a negative sign belongs, which is usually where marks are lost in a board exam setting.
A third area worth real attention is the equation of a plane in different forms: the normal form, the form through three points, and the form involving a line and a point. Many students try to memorise three separate formulas when really they only need to understand one idea properly: that a plane is defined by a point on it and a vector normal to it; everything else follows from that. Myclass24's approach in this chapter consistently returns to that single idea so that even an unfamiliar question in the exam does not feel unfamiliar.
Lastly, the angle between two lines, two planes, and a line and a plane all use the same underlying dot product logic, just applied to different vectors: direction ratios for lines and normal vectors for planes. Once a student sees these three cases laid out side by side rather than in isolation across different exercises, the chapter stops feeling like a list of unrelated formulas and starts feeling like one connected topic. That is the real goal of these solutions: not just correct answers, but a map of how the chapter holds together.