NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1
NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 contains all the questions with detailed solutions. Students are advised to solve these questions for better understanding of the concepts in exercise 11.1.
Krati Saraswat14 Feb, 2024
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NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 (Three-Dimensional Geometry)
NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 Three-Dimensional Geometry is prepared by academic team of Physics Wallah. We have prepared NCERT Solutions for all exercise of chapter 11. Given below is step by step solutions of all questions given in the NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 of Three Dimensional Geometry.NCERT Solutions for Class 12 Maths Chapter 11 Miscellaneous Exercise
NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 Overview
NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 covers these important topics. Students are encouraged to review each topic thoroughly in order to fully understand the concepts taught in the chapter and make optimal use of the provided solutions. These solutions are the outcome of the dedicated effort that the Physics Wallah teachers have been doing to aid students in understanding the ideas covered in this chapter. After going over and rehearsing these responses, the goal is for students to easily score outstanding exam results.NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1
Solve The Following Questions of NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1
Question 1.If a line makes angles 90°, 135°, 45° with x ,y and z axes respectively, find its direction cosines. Solution : A-line makes 90°, 135°, 45°with x, y and z axes respectively. Therefore, Direction cosines of the line are cos 90°, cos135°, and cos45° ⇒ Direction cosines of the line are 0,-1/√2 and 1/√2NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.2
Question 2.Find the direction cosines of a line which makes equal angles with the co-ordinate axes. Solution : Let the direction cosines of the line make an angle α with each of the coordinate axes. ∴ l = cos α , m = cos α , n = cos α
Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are
NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.3
Question 3.If a line has direction ratios −18, 12, −4, then what are its direction cosines? Solution : The direction ratios of the lines are -18, 12, -4 Direction cosines of the lines are
Hence, direction cosine of line are -9/11, 6/11 -2/11.
Question
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.
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)
Solution :
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
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
NCERT Solutions For Class 12 Maths Chapter 11 Exercise 11.1 FAQs
Why is Three-Dimensional Geometry important in mathematics?
Three-dimensional geometry plays a crucial role in various fields such as engineering, architecture, computer graphics, and physics. Understanding three-dimensional shapes and structures helps in visualizing and solving real-world problems accurately.
What is 3 dimensional geometry math?
In 3D geometry, we delve into the mathematics of shapes existing in three-dimensional space. This realm introduces three coordinates: the x-coordinate, y-coordinate, and z-coordinate. Within a 3D space, pinpointing the precise location of a point necessitates the consideration of three parameters.
What is the full form of 3D?
The full form of 3D is "Three-Dimensional."
How can I improve my problem-solving skills in Chapter 11?
Regular practice is essential for improving problem-solving skills. Work through the examples provided in the textbook, solve additional exercises, and attempt past examination questions. Seeking help from teachers or online resources for clarification on challenging topics can also be beneficial.
Why is 3d geometry important?
The 3d geometry helps in the representation of a line or a plane in a three-dimensional plane, using the x-axis, y-axis, z-axis.
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