# NCERT Exemplar Class 12 Maths Chapter 11 Solutions: Three Dimensional Geometry

NCERT Exemplar Class 12 Maths Solutions for Chapter 11 ‘Three Dimensional Geometry’ is designed for the students appearing in the CBSE Class 12 Board exam. Solving and practising these NCERT Exemplar Class 12 Maths problems will help you crack the CBSE Board as well as competitive exams such as JEE & NEET. This will add new knowledge to your previous base and help beat the competition in the CBSE Board exams.

In this Chapter of Three Dimensional Geometry, there are 49 exemplar problems divided into multiple exercises. You will encounter both objective and subjective questions. These NCERT Exemplar problems and solutions for Chapter 11 of CBSE Class 12 maths are specifically designed to make you think critically about three-dimensional geometry topics like direction cosines, straight lines, skew lines, and vector equations.

Instasolv’s NCERT Exemplar Solutions for all problems of Chapter 11 of Class 12 maths will prepare you for all levels of difficulties that are about to come in the CBSE exams. Our Solutions will help you in revising and implementing formulas of three-dimensional geometry efficiently.

## Important Topics for NCERT Exemplar Class 12 Maths Solutions Chapter 11 – Three Dimensional Geometry

NCERT Exemplar problems of Chapter 11 Class 12 maths include topics such as finding the angle formed between two lines and equations in vector forms. Some of the other significant issues to be covered are direction cosines and direction ratios, the equation of a plane, etc.

There are lots of theorems in this Chapter that are used solely to prove collinearity. There are other important topics, such as the distance between skew lines and parallel lines, the equation of a plane perpendicular to given vectors, or passing through three non-collinear points. There are discussions of the plane as well, and the distance or angle between 2 planes or a line to a plane.

**Direction cosines:** It is the cosines of the angles made by the line with positive directions of the coordinate axes. Direction cosines of a line joining two points P (x1, y1, z1) and Q (x2, y2, z2) are

**[x2-x1/PQ, y2-y1/PQ, z2-z1/PQ]**

**Direction ratios:** It is the numbers that are proportional to the direction cosines of the line.

**Skew lines: **These are lines in the space that are neither parallel nor intersecting. They lie in entirely different planes. The angle between skew lines is also the angle between two intersecting lines drawn from any point, which is parallel to each of the skew lines.

**Vector equation of a line:** This has got two cases

**Case 1**: equation of a line that passes through the given point whose position vector is ‘a’ and parallels to a given vector ‘b’ is

**r = a + λb**

**Case 2**: equation of a line that passes through two points whose positions vectors are ‘a’ and ‘b’ is

**r = a + λ(b – a)**

**Shortest distance:** The shortest distance between any two skew lines is the length of the line segment that is perpendicular to both those lines.

### Exercise Discussion of NCERT Exemplar Class 12 Maths Solutions Chapter 11 – Three Dimensional Geometry

- The first 15 exemplar questions are short answer types. They are focussed on finding the position vector, the vector equation, the angle between the lines, and finding the equation of the plane.
- The next 13 questions from 16-28 are long answer types, which are based mostly on numerical problems. They are significant mathematical problems based on calculating the equation of the plane, finding the foot of perpendicular and distance between the lines.
- 8 MCQs from 29-36 are asked where you have to choose one option; they are based on all the topics of the Chapter and mainly focus on finding the mirror image, locus of points, and direction cosines.
- From questions 37-41 are filling in the blanks problems require one-word answers. The next set of questions includes the true-false questions. It consists of the concepts that are commonly mistaken in exams like finding the value of the plane, the intercepts, and vectors with angles.

## Why Use NCERT Exemplar Class 12 Maths Solutions Chapter 11 by Instasolv?

- NCERT Exemplar Class 12 Maths Solutions for Chapter 11 is an original source of learning. Learning from here will always help you out in exams of all kinds.
- Instasolv NCERT Exemplar problems and solutions for Chapter 11 can be availed free of cost by any student.
- The solutions are stepwise, detailed and easy to understand. We have incorporated all the important concepts of the chapter.
- Our solutions will resolve all the problems that may arise in this chapter and will give you clarity regarding the concepts.