NCERT Solutions for Class 12 Maths Chapter 11: Three Dimensional Geometry - Exercise 11.1

1. If a line makes angles \[90^\circ ,135^\circ ,45^\circ \] with the \[x,y{\text{ }}\]and \[z\]axis respectively, find its direction cosines.

Ans: Let the direction of cosines of the given line be \[l,m\] and \[n\] .

Therefore,

\[l = \cos 90^\circ \]

\[l = 0\]

\[m = \cos 135^\circ \]

\[m =  - \dfrac{1}{{\sqrt 2 }}\]

\[n = \cos 45^\circ \]

\[n = \dfrac{1}{{\sqrt 2 }}\]

Therefore, the direction of cosines are \[0, - \dfrac{1}{{\sqrt 2 }}\] and \[\dfrac{1}{{\sqrt 2 }}\] .

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

Ans: Let the direction of the line that makes an angle \[\alpha \] with each of the coordinate axes.

Therefore,

\[l = \cos \alpha \]

\[m = \cos \alpha \]

\[n = \cos \alpha \]

As, we know that,

\[{l^2} + {m^2} + {n^2} = 1\]

So,

\[{\cos ^2}\alpha  + {\cos ^2}\alpha  + {\cos ^2}\alpha  = 1\]

\[3{\cos ^2}\alpha  = 1\]

\[{\cos ^2}\alpha  = \dfrac{1}{3}\]

\[\cos \alpha  =  \pm \dfrac{1}{{\sqrt 3 }}\]

Therefore, the direction of cosines of the line, which is equally inclined to the coordinate axes are \[ \pm \dfrac{1}{{\sqrt 3 }}, \pm \dfrac{1}{{\sqrt 3 }}\] and \[ \pm \dfrac{1}{{\sqrt 3 }}\].

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

Ans: In this question it is given the direction ratio a, b and c which is \[ - 18,12\]and \[ - 4\] respectively.

So,

\[a =  - 18\]

\[b = 12\]

\[c =  - 4\]

The direction cosines is given as,

\[l = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[l = \dfrac{{ - 18}}{{\sqrt {{{\left( { - 18} \right)}^2} + {{\left( {12} \right)}^2} + {{\left( { - 4} \right)}^2}} }}\]

\[l =  - \dfrac{{18}}{{22}}\]

\[m = \dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[m = \dfrac{{12}}{{\sqrt {{{\left( { - 18} \right)}^2} + {{\left( {12} \right)}^2} + {{\left( { - 4} \right)}^2}} }}\]

\[m = \dfrac{{12}}{{22}}\]

\[n = \dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[n = \dfrac{{ - 4}}{{\sqrt {{{\left( { - 18} \right)}^2} + {{\left( {12} \right)}^2} + {{\left( { - 4} \right)}^2}} }}\]

\[n =  - \dfrac{2}{{22}}\]

Therefore, the direction cosines are \[ - \dfrac{{18}}{{22}},\dfrac{{12}}{{22}}\] and \[ - \dfrac{2}{{22}}\].

4. Show that the points \[\left( {2,3,4} \right),\left( { - 1, - 2,1} \right),\left( {5,8,7} \right)\] are collinear.

Ans: The given points are \[A\left( {2,3,4} \right),B\left( { - 1, - 2,1} \right)\] and \[C\left( {5,8,7} \right)\] .

The direction ratio of line joining the two coordinates \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given as \[\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right)\] and \[\left( {{z_1} - {z_2}} \right)\] .

The direction ratio of line \[AB\] is given as \[\left( { - 1 - 2} \right),\left( { - 2 - 3} \right)\] and \[\left( {1 - 4} \right)\] .

So, the direction ratio of line \[AB\] is \[ - 3, - 5\] and \[ - 3\] .

The direction ratio of line \[BC\] is given as \[\left( {5 - \left( { - 1} \right)} \right),\left( {8 - \left( { - 2} \right)} \right)\] and \[\left( {7 - 1} \right)\] .

So, the direction ratio of line \[BC\] is 6, 10 and 6 .

On comparing the direction ratio of \[AB\] and \[BC\], it can be seen that the direction ratio of \[BC\] is \[ - 2\] times of \[AB\] i.e. they are proportional.

\[AB = \lambda \left( {BC} \right)\]

Therefore, \[AB\parallel BC\]. As point B is common to both \[AB\] and \[BC\] .

Therefore, given points \[A\left( {2,3,4} \right),B\left( { - 1, - 2,1} \right)\] and \[C\left( {5,8,7} \right)\] 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).

Ans: The given vertices of \[\Delta ABC\] are \[A\left( {3,5, - 4} \right),B\left( { - 1,1,2} \right)\] and \[C\left( { - 5, - 5, - 2} \right)\] .

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Calculating direction cosines of side \[AB\].

The direction ratio of line joining the two coordinates \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given as \[\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right)\] and \[\left( {{z_1} - {z_2}} \right)\] .

The direction ratio of \[AB\] is given as \[\left( { - 1 - 3} \right),\left( {1 - 5} \right)\] and \[\left( {2 - \left( { - 4} \right)} \right)\] .

The direction ratio of \[AB\] is \[ - 4, - 4\] and \[6\] .

The direction cosines of side \[AB\] using direction ratio is given as,

\[l = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[l = \dfrac{{ - 4}}{{\sqrt {{{\left( { - 4} \right)}^2} + {{\left( { - 4} \right)}^2} + {{\left( 6 \right)}^2}} }}\]

\[l =  - \dfrac{4}{{2\sqrt {17} }}\]

\[l =  - \dfrac{2}{{\sqrt {17} }}\]

\[m = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[m = \dfrac{{ - 4}}{{\sqrt {{{\left( { - 4} \right)}^2} + {{\left( { - 4} \right)}^2} + {{\left( 6 \right)}^2}} }}\]

\[m =  - \dfrac{4}{{2\sqrt {17} }}\]

\[m =  - \dfrac{2}{{\sqrt {17} }}\]

\[n = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[n = \dfrac{6}{{\sqrt {{{\left( { - 4} \right)}^2} + {{\left( { - 4} \right)}^2} + {{\left( 6 \right)}^2}} }}\]

\[n = \dfrac{6}{{2\sqrt {17} }}\]

\[n = \dfrac{3}{{\sqrt {17} }}\]

So, the direction cosines of \[AB\] is \[ - \dfrac{2}{{\sqrt {17} }}, - \dfrac{2}{{\sqrt {17} }}\] and \[\dfrac{3}{{\sqrt {17} }}\] .

Calculating the direction cosines of side \[BC\]

The direction ratio of line joining the two coordinates \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given as \[\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right)\] and \[\left( {{z_1} - {z_2}} \right)\] .

The direction ratio of \[BC\] is given as \[\left( { - 1 - 3} \right),\left( {1 - 5} \right)\] and \[\left( {2 - \left( { - 4} \right)} \right)\] .

The direction ratio of \[BC\]  is \[ - 4, - 4\] and \[6\] .

The direction cosines of side \[BC\] using direction ratio is given as,

\[l = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[l = \dfrac{{ - 4}}{{\sqrt {{{\left( { - 4} \right)}^2} + {{\left( { - 6} \right)}^2} + {{\left( { - 4} \right)}^2}} }}\]

\[l =  - \dfrac{4}{{2\sqrt {17} }}\]

\[l =  - \dfrac{2}{{\sqrt {17} }}\]

\[m = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[m = \dfrac{{ - 6}}{{\sqrt {{{\left( { - 4} \right)}^2} + {{\left( { - 6} \right)}^2} + {{\left( { - 4} \right)}^2}} }}\]

\[m =  - \dfrac{6}{{2\sqrt {17} }}\]

\[m =  - \dfrac{3}{{\sqrt {17} }}\]

\[n = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[n = \dfrac{{ - 4}}{{\sqrt {{{\left( { - 4} \right)}^2} + {{\left( { - 6} \right)}^2} + {{\left( { - 4} \right)}^2}} }}\]

\[n =  - \dfrac{4}{{2\sqrt {17} }}\]

\[n =  - \dfrac{2}{{\sqrt {17} }}\]

So, the direction cosines of \[BC\] is \[ - \dfrac{2}{{\sqrt {17} }}, - \dfrac{3}{{\sqrt {17} }}\] and \[ - \dfrac{2}{{\sqrt {17} }}\] .

Calculating the direction cosines of side \[AC\]

The direction ratio of line joining the two coordinates \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given as \[\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right)\] and \[\left( {{z_1} - {z_2}} \right)\] .

The direction ratio of \[AC\] is given as \[\left( { - 5 - 3} \right),\left( { - 5 - 5} \right)\] and \[\left( { - 2 - \left( { - 4} \right)} \right)\] .

The direction ratio of \[AC\]  is \[ - 8, - 10\] and 2 .

The direction cosines of side \[AC\] using direction ratio is given as,

\[l = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[l = \dfrac{{ - 8}}{{\sqrt {{{\left( { - 8} \right)}^2} + {{\left( { - 10} \right)}^2} + {{\left( 2 \right)}^2}} }}\]

\[l =  - \dfrac{8}{{2\sqrt {42} }}\]

\[l =  - \dfrac{4}{{\sqrt {42} }}\]

\[m = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[m = \dfrac{{ - 10}}{{\sqrt {{{\left( { - 8} \right)}^2} + {{\left( { - 10} \right)}^2} + {{\left( 2 \right)}^2}} }}\]

\[m =  - \dfrac{{10}}{{2\sqrt {42} }}\]

\[m =  - \dfrac{5}{{\sqrt {42} }}\]

\[n = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]

\[n = \dfrac{2}{{\sqrt {{{\left( { - 8} \right)}^2} + {{\left( { - 10} \right)}^2} + {{\left( 2 \right)}^2}} }}\]

\[n = \dfrac{2}{{2\sqrt {42} }}\]

\[n = \dfrac{1}{{\sqrt {42} }}\]

So, the direction cosines of \[BC\] is \[ - \dfrac{4}{{\sqrt {42} }}, - \dfrac{5}{{\sqrt {42} }}\] and \[\dfrac{1}{{\sqrt {42} }}\] .

Therefore,

Direction cosines of \[AB\] is \[ - \dfrac{2}{{\sqrt {17} }}, - \dfrac{2}{{\sqrt {17} }}\] and \[\dfrac{3}{{\sqrt {17} }}\] .

Direction cosines of \[BC\] is \[ - \dfrac{2}{{\sqrt {17} }}, - \dfrac{3}{{\sqrt {17} }}\] and \[ - \dfrac{2}{{\sqrt {17} }}\] .

Direction cosines of \[BC\] is \[ - \dfrac{4}{{\sqrt {42} }}, - \dfrac{5}{{\sqrt {42} }}\] and \[\dfrac{1}{{\sqrt {42} }}\] .

NCERT Solutions for Class 12 Maths PDF Download

• Chapter 1 - Relations and Functions

• Chapter 2 - Inverse Trigonometric Functions

• Chapter 3 - Matrices

• Chapter 4 - Determinants

• Chapter 5 - Continuity and Differentiability

• Chapter 6 - Application of Derivatives

• Chapter 7 - Integrals

• Chapter 8 - Application of Integrals

• Chapter 9 - Differential Equations

• Chapter 10 - Vector Algebra

• Chapter 11 - Three Dimensional Geometry

• Chapter 12 - Linear Programming

• Chapter 13 - Probability

NCERT Solution Class 12 Maths of Chapter 11 All Exercises

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.1

Class 12, Chapter 11 Three Dimensional Geometry discusses the fundamentals of 3-D geometry like the direction cosine and direct relationship of a line joining two points and also the equations of lines and planes in different conditions, the angle between two lines, and the angle between two planes. 

You will be learning the following things: 

• What are a line and a plane?

• The shortest distance between two oblique lines

• The distance of a plane from a point. Please note that most of the above results are obtained in vector form. 

Above all, we will also translate these results into Cartesian form, which sometimes presents a clear geometric and analytical picture of the situation.

Understanding Equation of a Line in Space

In Class 11, we have studied the equation of lines in two dimensions, now we will study vector and Cartesian equations of a line in space. 

A line is uniquely determined if 

• A line passes through a given point and its direction is given,

• A line passes through two given points.

Important of Class 12 Maths Exercise 11.1

There are 5 questions in exercise 11 that are based on simple concepts of 3-D geometry. 

You should do all these questions to understand the concept because these concepts will be helpful in solving questions in later exercises as well.

Opting for the NCERT solutions for Ex 11.1 Class 12 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 11.1 Class 12 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 12 students who are thorough with all the concepts from the Maths textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 12 Maths Chapter 11 Exercise 11.1 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.

Besides these NCERT solutions for Class 12 Maths Chapter 11 Exercise 11.1, there are plenty of exercises in this chapter which contain innumerable questions as well. All these questions are solved/answered by our in-house subject experts as mentioned earlier. Hence all of these are bound to be of superior quality and anyone can refer to these during the time of exam preparation. In order to score the best possible marks in the class, it is really important to understand all the concepts of the textbooks and solve the problems from the exercises given next to it. 

Do not delay any more. Download the NCERT solutions for Class 12 Maths Chapter 11 Exercise 11.1 from Vedantu website now for better exam preparation. If you have the Vedantu app in your phone, you can download the same through the app as well. The best part of these solutions is these can be accessed both online and offline as well.