NCERT Solutions for Maths Class 12 Chapter 11 Exercise 11.1 - FREE PDF Download
Exercise 11.1 specifically focuses on the Direction Cosines and Direction Ratios of a line. Ex 11.1 Class 12 introduces students to the concept of three-dimensional space, where they learn to describe the orientation of lines using direction cosines and ratios. Direction cosines are the cosines of the angles that a line makes with the positive directions of the coordinate axes. Students can download Class 12 Maths NCERT Solutions from our page which is prepared so that you can understand it easily.
- 4.1Exercise 11.1
Class 12 Maths Ex 11.1 Solutions are aligned with the updated CBSE guidelines for Class 12, ensuring students are well-prepared for exams. Access the Class 12 Maths Syllabus here.
Glance on NCERT Solutions Maths Chapter 11 Exercise 11.1 Class 12 | Vedantu
NCERT Solutions for Class 12 Maths Ex 11.1 covers the topic of Direction Cosines and Direction Ratios of a Line.
Direction Cosines of a Line: Direction cosines are the cosines of the angles that a line makes with the positive directions of the coordinate axes (x, y, and z). They are denoted as l, m, and n, representing the cosines of the angles between the line and the x-axis, y-axis, and z-axis, respectively. These values help in understanding the orientation of a line in three-dimensional space.
Direction Ratios of a Line: Direction ratios are a set of three numbers that are proportional to the direction cosines.
Ex 11.1 Class 12 covers 5 fully solved questions and solutions.
Formulas Used in Class 12 Chapter 11 Exercise 11.1
1. Direction Cosines in Class 12 Ex 11.1 is Defined as:
These represent the cosines of the angles a line makes with the positive x, y, and z axes. Let l, m, and n be the direction cosines of a line.
Relationship between direction cosines: l² + m² + n² = 1 (This ensures the cosines correspond to a valid direction vector)
2. Direction Ratios in Class 12 Ex 11.1 is Defined as:
These represent the ratios of the components along the x, y, and z axes for a line's direction vector. If (a, b, c) are the direction ratios, they are proportional to the direction cosines (l, m, n).
Relationship between direction cosines and ratios: a/l = b/m = c/n
3. Finding Direction Cosines:
Given the direction ratios (a, b, c): l = a/√(a² + b² + c²), m = b/√(a² + b² + c²), n = c/√(a² + b² + c²)
Given the angles made with the axes (α, β, γ): l = cos(α), m = cos(β), n = cos(γ) (Though these angles might not be directly given in the exercise, the concepts might be used)
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 \[C\left( { - 5, - 5, - 2} \right)\] .
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 \[\dfrac{1}{{\sqrt {42} }}\] .
Conclusion
In Ex 11.1 Class 12 on Three Dimensional Geometry, students explore the fundamental concepts of Direction Cosines and Direction Ratios of a line. These concepts are essential for understanding the orientation and direction of lines in three-dimensional space. By understanding the use of direction cosines and ratios, students can accurately describe and analyze the spatial relationships of lines. Class 12 Ex 11.1 provides a solid foundation for more advanced topics in three-dimensional geometry and its applications in fields such as physics, engineering, and computer graphics. Through this exercise, students develop critical problem-solving skills and gain confidence in working with three-dimensional geometric concepts.
Class 12 Maths Chapter 11: Exercises Breakdown
CBSE Class 12 Maths Chapter 11 Other Study Materials
NCERT Solutions for Class 12 Maths | Chapter-wise List
Given below are the chapter-wise NCERT 12 Maths solutions PDF. Using these chapter-wise class 12th maths ncert solutions, you can get clear understanding of the concepts from all chapters.
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FAQs on NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.1
1. What is the weightage of Exercise 11.1 of class 12 Maths?
The weightage of any exercise is decided randomly based on the chapter and the format. For board exams, the sums of any chapter can be randomly selected and hence, it is important to practice all the sums to be ready for any question that is set. Apart from this students are advised to check the chapter-wise weightage to get a clearer picture.
2. Which question is considered an important one from Class 12 Maths exercise 11.1?
No such thing as the most significant question exists! You see, the easier ones help you remember your principles, while the more complex ones evaluate your problem-solving abilities and how you apply your knowledge in practice. To understand the topics in Chapter 11 3D Geometry you need to pay close attention. As a result, please practice all amounts, no matter how tough they are. You can visit the page NCERT Solutions Class 12 Maths Chapter 11 on the official website of Vedantu.
3. What exactly is 3D Geometry?
Shapes in 3D space may be calculated using 3 coordinates on the XYZ plane: the x-coordinate, y-coordinate, and z-coordinate. Three-dimensional forms are those that occupy space. As a solid form with three dimensions (length, width, and height), 3D shapes may also be described in this way. It is an integral part of Mathematics and students must learn its concepts very clearly and practice all the problems related to the same for better understanding.
4. What is the need for 3D Geometry?
As the name suggests, 3D geometry is the study of three-dimensional forms in space. A z-coordinate may be determined by subtracting the x-coordinate from the value of the y-coordinate. If you want to discover the precise position of a point in three-dimensional space, you need three parameters. It is a vast field and also has multiple applications in the domain of mathematics.
5. Where can we apply Geometry?
A few examples of how geometry is used in the actual world include computer-aided design (CAD) for construction plans and industrial assembly systems. These include nanotechnology, computer graphics, visual graphs, video game programming and virtual reality development (VR). These are just a few examples of many of Geometry’s applications in our life and the domain of Mathematics.
6. What are direction cosines in Ex 11.1 Class 12 Maths?
Direction cosines are the cosines of the angles that a line makes with the positive directions of the coordinate axes. They are denoted by l, m, and n for the x, y, and z axes, respectively.
7. What is the significance of direction cosines in Ex 11.1 Class 12 Maths?
Direction cosines provide a way to quantify the orientation of a line in three-dimensional space. They are essential for understanding and solving problems related to the spatial orientation of lines.
8. What kind of problems can be solved using direction cosines and ratios?
Problems involving the orientation and angles between lines in three-dimensional space, the equation of a line, and spatial relationships can be solved using direction cosines and ratios. Practise more problems provided by Vedantu in Class 12 Maths Ch 11 Ex 11.1.
9. How many questions from Class 12 Maths Ch 11 Ex 11.1 typically appear in exams?
The number of questions from Exercise 11.1 in exams can vary generally 1-2 questions have appeared in previous years' exams, but typically, questions on direction cosines and direction ratios are common. Understanding these concepts thoroughly will help in tackling related problems in competitive and board exams.
