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# NCERT Solution - 3D Geometry Class 11 Notes | EduRev

## Class 11 : NCERT Solution - 3D Geometry Class 11 Notes | EduRev

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Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 1 of 17
Exercise 12.1
Question 1:
A point is on the x-axis. What are its y-coordinates and z-coordinates?
If a point is on the x-axis, then its y-coordinates and z-coordinates are zero.

Question 2:
A point is in the XZ-plane. What can you say about its y-coordinate?
If a point is in the XZ plane, then its y-coordinate is zero.

Question 3:
Name the octants in which the following points lie:
(1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5),
(â€“3, â€“1, 6), (2, â€“4, â€“7)
The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive.
Therefore, this point lies in octant I.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive,
negative, and positive respectively. Therefore, this point lies in octant IV.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.
The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative,
positive, and positive respectively. Therefore, this point lies in octant II.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative,
negative, and positive respectively. Therefore, this point lies in octant III.
The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Page 2

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 1 of 17
Exercise 12.1
Question 1:
A point is on the x-axis. What are its y-coordinates and z-coordinates?
If a point is on the x-axis, then its y-coordinates and z-coordinates are zero.

Question 2:
A point is in the XZ-plane. What can you say about its y-coordinate?
If a point is in the XZ plane, then its y-coordinate is zero.

Question 3:
Name the octants in which the following points lie:
(1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5),
(â€“3, â€“1, 6), (2, â€“4, â€“7)
The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive.
Therefore, this point lies in octant I.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive,
negative, and positive respectively. Therefore, this point lies in octant IV.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.
The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative,
positive, and positive respectively. Therefore, this point lies in octant II.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative,
negative, and positive respectively. Therefore, this point lies in octant III.
The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 2 of 17
Question 4:
Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as .
(ii) The coordinates of points in the XY-plane are of the form .
(iii) Coordinate planes divide the space into octants.

Page 3

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 1 of 17
Exercise 12.1
Question 1:
A point is on the x-axis. What are its y-coordinates and z-coordinates?
If a point is on the x-axis, then its y-coordinates and z-coordinates are zero.

Question 2:
A point is in the XZ-plane. What can you say about its y-coordinate?
If a point is in the XZ plane, then its y-coordinate is zero.

Question 3:
Name the octants in which the following points lie:
(1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5),
(â€“3, â€“1, 6), (2, â€“4, â€“7)
The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive.
Therefore, this point lies in octant I.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive,
negative, and positive respectively. Therefore, this point lies in octant IV.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.
The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative,
positive, and positive respectively. Therefore, this point lies in octant II.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative,
negative, and positive respectively. Therefore, this point lies in octant III.
The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 2 of 17
Question 4:
Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as .
(ii) The coordinates of points in the XY-plane are of the form .
(iii) Coordinate planes divide the space into octants.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 3 of 17
Exercise 12.2
Question 1:
Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1) (ii) (â€“3, 7, 2) and (2, 4, â€“1)
(iii) (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) (2, â€“1, 3) and (â€“2, 1, 3)
The distance between points P(x
1
, y
1
, z
1
) and P(x
2
, y
2
, z
2
) is given by

(i) Distance between points (2, 3, 5) and (4, 3, 1)

(ii) Distance between points (â€“3, 7, 2) and (2, 4, â€“1)

(iii) Distance between points (â€“1, 3, â€“4) and (1, â€“3, 4)

(iv) Distance between points (2, â€“1, 3) and (â€“2, 1, 3)
Page 4

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 1 of 17
Exercise 12.1
Question 1:
A point is on the x-axis. What are its y-coordinates and z-coordinates?
If a point is on the x-axis, then its y-coordinates and z-coordinates are zero.

Question 2:
A point is in the XZ-plane. What can you say about its y-coordinate?
If a point is in the XZ plane, then its y-coordinate is zero.

Question 3:
Name the octants in which the following points lie:
(1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5),
(â€“3, â€“1, 6), (2, â€“4, â€“7)
The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive.
Therefore, this point lies in octant I.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive,
negative, and positive respectively. Therefore, this point lies in octant IV.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.
The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative,
positive, and positive respectively. Therefore, this point lies in octant II.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative,
negative, and positive respectively. Therefore, this point lies in octant III.
The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 2 of 17
Question 4:
Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as .
(ii) The coordinates of points in the XY-plane are of the form .
(iii) Coordinate planes divide the space into octants.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 3 of 17
Exercise 12.2
Question 1:
Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1) (ii) (â€“3, 7, 2) and (2, 4, â€“1)
(iii) (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) (2, â€“1, 3) and (â€“2, 1, 3)
The distance between points P(x
1
, y
1
, z
1
) and P(x
2
, y
2
, z
2
) is given by

(i) Distance between points (2, 3, 5) and (4, 3, 1)

(ii) Distance between points (â€“3, 7, 2) and (2, 4, â€“1)

(iii) Distance between points (â€“1, 3, â€“4) and (1, â€“3, 4)

(iv) Distance between points (2, â€“1, 3) and (â€“2, 1, 3)

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 4 of 17

Question 2:
Show that the points (â€“2, 3, 5), (1, 2, 3) and (7, 0, â€“1) are collinear.
Let points (â€“2, 3, 5), (1, 2, 3), and (7, 0, â€“1) be denoted by P, Q, and R respectively.
Points P, Q, and R are collinear if they lie on a line.

Page 5

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 1 of 17
Exercise 12.1
Question 1:
A point is on the x-axis. What are its y-coordinates and z-coordinates?
If a point is on the x-axis, then its y-coordinates and z-coordinates are zero.

Question 2:
A point is in the XZ-plane. What can you say about its y-coordinate?
If a point is in the XZ plane, then its y-coordinate is zero.

Question 3:
Name the octants in which the following points lie:
(1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5),
(â€“3, â€“1, 6), (2, â€“4, â€“7)
The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive.
Therefore, this point lies in octant I.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive,
negative, and positive respectively. Therefore, this point lies in octant IV.
The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.
The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative,
positive, and positive respectively. Therefore, this point lies in octant II.
The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative,
negative, and positive respectively. Therefore, this point lies in octant III.
The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 2 of 17
Question 4:
Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as .
(ii) The coordinates of points in the XY-plane are of the form .
(iii) Coordinate planes divide the space into octants.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 3 of 17
Exercise 12.2
Question 1:
Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1) (ii) (â€“3, 7, 2) and (2, 4, â€“1)
(iii) (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) (2, â€“1, 3) and (â€“2, 1, 3)
The distance between points P(x
1
, y
1
, z
1
) and P(x
2
, y
2
, z
2
) is given by

(i) Distance between points (2, 3, 5) and (4, 3, 1)

(ii) Distance between points (â€“3, 7, 2) and (2, 4, â€“1)

(iii) Distance between points (â€“1, 3, â€“4) and (1, â€“3, 4)

(iv) Distance between points (2, â€“1, 3) and (â€“2, 1, 3)

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 4 of 17

Question 2:
Show that the points (â€“2, 3, 5), (1, 2, 3) and (7, 0, â€“1) are collinear.
Let points (â€“2, 3, 5), (1, 2, 3), and (7, 0, â€“1) be denoted by P, Q, and R respectively.
Points P, Q, and R are collinear if they lie on a line.

Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths

Page 5 of 17

Here, PQ + QR = PR
Hence, points P(â€“2, 3, 5), Q(1, 2, 3), and R(7, 0, â€“1) are collinear.

Question 3:
Verify the following:
(i) (0, 7, â€“10), (1, 6, â€“6) and (4, 9, â€“6) are the vertices of an isosceles triangle.
(ii) (0, 7, 10), (â€“1, 6, 6) and (â€“4, 9, 6) are the vertices of a right angled triangle.
(iii) (â€“1, 2, 1), (1, â€“2, 5), (4, â€“7, 8) and (2, â€“3, 4) are the vertices of a parallelogram.
(i) Let points (0, 7, â€“10), (1, 6, â€“6), and (4, 9, â€“6) be denoted by A, B, and C
respectively.

Here, AB = BC ? CA
Thus, the given points are the vertices of an isosceles triangle.
(ii) Let (0, 7, 10), (â€“1, 6, 6), and (â€“4, 9, 6) be denoted by A, B, and C respectively.
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