NCERT Solutions Class 12 Mathematics Chapter 11 Three Dimensional Geometry have been provided below and is also available in Pdf for free download. The NCERT solutions for Class 12 Mathematics have been prepared as per the latest syllabus, NCERT books and examination pattern suggested in Class 12 by CBSE, NCERT and KVS. Questions given in NCERT book for Class 12 Mathematics are an important part of exams for Class 12 Mathematics and if answered properly can help you to get higher marks. Refer to more Chapter-wise answers for NCERT Class 12 Mathematics and also download more latest study material for all subjects. Chapter 11 Three Dimensional Geometry is an important topic in Class 12, please refer to answers provided below to help you score better in exams

## Chapter 11 Three Dimensional Geometry Class 12 Mathematics NCERT Solutions

Class 12 Mathematics students should refer to the following NCERT questions with answers for Chapter 11 Three Dimensional Geometry in Class 12. These NCERT Solutions with answers for Class 12 Mathematics will come in exams and help you to score good marks

### Chapter 11 Three Dimensional Geometry NCERT Solutions Class 12 Mathematics

**Exercise 11.1 **

**1. If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.**

**Solution**

Let direction cosines of the line be l, m, and n.

I = cos 90° = 0

m = cos 135° = -1/√2

n = cos 45° = 1/√2

Therefore, the direction cosines of the line are 0, -1/√2 and 1/√2.

**2. Find the direction cosines of a line which makes equal angles with the coordinate axes.**

**Solution**

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

∴ I = cos α, m = cos α, n = cos α

l^{2} + m^{2} + n^{2} = 1

⇒ cos^{2} α + cos^{2 }α + cos^{2} α = 1

⇒ 3 cos^{2} α = 1

⇒ cos^{2} α = 1/3

⇒ cos α = ± 1/√3

Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are ± 1/√3, ± 1/√3, and ± 1/√3.

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

**Solution**

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

i.e., -18/22 , 12/22, -4/22

-9/11, 6/11, -2/11

Thus, the direction cosines are -9/11, 6/11, and -2/11.

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

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 line joining the points, (

*x*

_{1},

*y*

_{1},

*z*

_{1}) and (

*x*

_{2},

*y*

_{2},

*z*

_{2}), are given by,

*x*

_{2}−

*x*

_{1},

*y*

_{2}−

*y*

_{1}, and

*z*

_{2}−

*z*

_{1}.

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 are collinear.

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

The vertices of Δ ABC are A(3, 5, -4), B(-1, 1, 2) and C(-5, -5, -2).

-217, -317, -217The 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 882 + 102 +(-22), 1082 + 102 + -22, -282 + 102 + (-228242), 10242, -2242442, 542, -142

NCERT Solutions Class 12 Mathematics Chapter 1 Relations and Functions |

NCERT Solutions Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions |

NCERT Solutions Class 12 Mathematics Chapter 3 Matrices |

NCERT Solutions Class 12 Mathematics Chapter 4 Determinants |

NCERT Solutions Class 12 Mathematics Chapter 5 Continuity and Differentiability |

NCERT Solutions Class 12 Mathematics Chapter 6 Application of Derivatives |

NCERT Solutions Class 12 Mathematics Chapter 7 Integrals |

NCERT Solutions Class 12 Mathematics Chapter 8 Application of Integrals |

NCERT Solutions Class 12 Mathematics Chapter 9 Differential Equations |

NCERT Solutions Class 12 Mathematics Chapter 10 Vector Algebra |

NCERT Solutions Class 12 Mathematics Chapter 11 Three Dimensional Geometry |

NCERT Solutions Class 12 Mathematics Chapter 12 Linear Programming |

NCERT Solutions Class 12 Mathematics Chapter 13 Probability |

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### NCERT Solutions Class 12 Mathematics Chapter 11 Three Dimensional Geometry

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### Chapter 11 Three Dimensional Geometry Class 12 Mathematics NCERT Solutions

The Class 12 Mathematics NCERT Solutions Chapter 11 Three Dimensional Geometry are designed in a way that will help to improve the overall understanding of students. The answers to each question in Chapter 11 Three Dimensional Geometry of Mathematics Class 12 has been designed based on the latest syllabus released for the current year. We have also provided detailed explanations for all difficult topics in Chapter 11 Three Dimensional Geometry Class 12 chapter of Mathematics so that it can be easier for students to understand all answers.

**NCERT Solutions Chapter 11 Three Dimensional Geometry Class 12 Mathematics**

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**Chapter 11 Three Dimensional Geometry Class 12 NCERT Solution Mathematics**

These solutions of Chapter 11 Three Dimensional Geometry NCERT Questions given in your textbook for Class 12 Mathematics have been designed to help students understand the difficult topics of Mathematics in an easy manner. These will also help to build a strong foundation in the Mathematics. There is a combination of theoretical and practical questions relating to all chapters in Mathematics to check the overall learning of the students of Class 12.

**Class 12 NCERT Solution Mathematics Chapter 11 Three Dimensional Geometry**

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