# Three-Dimensional Geometry Chapter 11 Class 12 Maths NCERT Textbook PDF

NCERT Solutions for Class 12 Maths Chapter 11‘ PDF Quick download link is given at the bottom of this article. You can see the PDF demo, size of the PDF, page numbers, and direct download Free PDF of ‘Ncert Class 12 Maths Chapter 11 Exercise Solution’ using the download button.

### Chapter 11: Three-dimensional(3D) Geometry

#### 11.1 Introduction

In Class XI, while studying Analytical Geometry in two dimensions, and the introduction to three-dimensional geometry, we are confined to the Cartesian methods only.

In the previous chapter of this book, we have studied some basic concepts of vectors. We will now use vector algebra for three-dimensional geometry.

The purpose of this approach to 3-dimensional geometry is that it makes the study simple and elegant*.

In this chapter, we shall study the direction of cosines and direction ratios of a line joining two points and also discuss the equations of lines and planes in space under different conditions, the angle between two lines, two planes, a line, and a plane, the shortest distance between two skew lines, and the distance of a point from a plane.

Most of the above results are obtained in vector form. Nevertheless, we shall also translate these results into the Cartesian form which, at times, presents a more clear geometric and analytic picture of the situation

### NCERT Solutions Class 12 Maths Chapter 11 Three-dimensional Geometry

Question 1:

If a line makes angles 90°, 135°, and 45° with xy, and z-axes respectively, find its direction cosines.

Let direction cosines of the line be lm, and n.

Therefore, the direction cosines of the line are Question 2:

Find the direction cosines of a line that makes equal angles with the coordinate axes.

Let the direction cosines of the line make an angle α with each of the coordinate axes.

∴ l = cos αm = cos αn = cos α

Thus, the direction cosines of the line, which are equally inclined to the coordinate axes, are Question 3:

If a line has the direction ratios −18, 12, and −4, then what are its direction cosines?

If a line has direction ratios of −18, 12, and −4, then its direction cosines are

Thus, the direction cosines are .

Question 4:

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

The given points are A (2, 3, 4), B (− 1, − 2, 1), and C (5, 8, 7).

It is known that the direction ratios of a line joining the points, (x1y1z1) and (x2y2z2), are given by, x2 − x1y2 − y1, and z2 − z1.

The direction ratios of AB are (−1 − 2), (−2 − 3), and (1 − 4) i.e., −3, −5, and −3.

The direction ratios of BC are (5 − (− 1)), (8 − (− 2)), and (7 − 1) i.e., 6, 10, and 6.

It can be seen that the direction ratios of BC are −2 times that of AB i.e., they are proportional.

Therefore, AB is parallel to BC. Since point B is common to both AB and BC, points A, B, and C is collinear.

Question 5:

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, − 4), (− 1, 1, 2), and (− 5, − 5, − 2)

The vertices of ΔABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2).

The direction ratios of side AB are (−1 − 3), (1 − 5), and (2 − (−4)) i.e., −4, −4, and 6.

Therefore, the direction cosines of AB are

The direction ratios of BC are (−5 − (−1)), (−5 − 1), and (−2 − 2) i.e., −4, −6, and −4.

Therefore, the direction cosines of BC are −217√, −317√, −217√-217, -317, -217
The direction ratios of CA are 3−(−5), 5−(−5), and −4−(−2) i.e. 8, 10, and -2.

Therefore the direction cosines of CA are

8(8)2 + (10)2 + (−2)2√, 10(8)2 + (10)2 + (−2)2√, −2(8)2 + (10)2 + (−2)2√8242√, 10242√, −2242√442√, 542√, −142√