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NCERT Solutions for Class 12 Maths, Chapter 11 - Three Dimensional Geometry

Last year, you studied analytical Geometry in both two and three dimensions (only
Cartesian Methods). You also studied basic terms related to vectors in the previous
chapter. In NCERT Class 12 Maths, Chapter 11, both these topics will be merged and
you’ll learn how to use vector algebra to three-dimensional geometry. In this chapter,
you’ll learn many concepts related to three-dimensional geometry like the direction
cosines and direction ratios of a line joining two points, equation of a line and plane in
space under various conditions, the angle between two lines, the shortest distance
between two lines and more. As everything around us is three dimensional in shape,
getting an in-depth knowledge of three-dimensional geometry will lay a strong
foundation in your professional career ahead.

Let’s take a look at topics and sub-topics of Class 12 Maths, Chapter 11, Three Dimensional Geometry-

11.1 Introduction
11.2 Direction Cosines and Direction Ratios of a Line
11.3 Equation of a Line in Space
11.4 Angle between Two Lines
11.5 Shortest Distance between Two Lines
11.6 Plane
11.7 Coplanarity of Two Lines
11.8 Angle between Two Planes
11.9 Distance of a Point from a Plane
11.10 Angle between a Line and a Plane

In Class 12 Maths Chapter 11, you will learn – 

Section 11.2 – In this section, you’ll learn about direction cosines and direction ratios of
a line, the relation between the direction cosines of a line, direction cosines of a line
passing through two points and more.

Section 11.3 – In this section, you’ll study the equation of a line through a given point
and parallel to a given vector. You will acquire knowledge about the equation of a line
passing through two given points, derivation of cartesian form from vector form and

Section 11.4 – In this section, you’ll be able to find the angle between two lines when
their equations are given. You’ll also learn how to find the angle between the pair of

Section 11.5 – In this section, you’ll study the shortest distance between two lines.

Section 11.6 – In this section, you’ll learn terms such as the equation of a plane in normal
form, equation of a plane perpendicular to a given vector and passing through a given
point, equation of a plane passing through three non-collinear points, intercept form of
the equation of a plane, a plane passing through the intersection of two given planes and

Section 11.7 – In this section learn what are coplanar lines and in which situations the
lines are coplanar.

Section 11.8 – In this section, you will come to know that the angle between two planes
is defined as the angle between their normals.

Section 11.9 – In this section, you will learn about the Distance of a Point from a Plane.

Section 11.10 – In this section, you’ll learn that the angle between a line and a plane is
the complement of the angle between the line and normal to the plane.

To know more about the Class 12 Maths Chapter 11 Three Dimensional Geometry,
you should explore the exercises below. You must even download the Three
Dimensional Geometry Class 12 NCERT Solutions PDF, solved by expert Maths

Also, solve miscellaneous exercises of Class 12 Maths Chapter 11, Three Dimensional

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