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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 – 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, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –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, –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, –2, –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, –5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (–4, 2, –5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (–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 (–3, –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, –4, –7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Page 2

Class XI Chapter 12 – 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, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –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, –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, –2, –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, –5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (–4, 2, –5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (–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 (–3, –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, –4, –7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 – 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 – 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, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –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, –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, –2, –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, –5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (–4, 2, –5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (–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 (–3, –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, –4, –7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 – 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 – 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) (–3, 7, 2) and (2, 4, –1)
(iii) (–1, 3, –4) and (1, –3, 4) (iv) (2, –1, 3) and (–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 (–3, 7, 2) and (2, 4, –1)

(iii) Distance between points (–1, 3, –4) and (1, –3, 4)

(iv) Distance between points (2, –1, 3) and (–2, 1, 3)
Page 4

Class XI Chapter 12 – 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, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –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, –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, –2, –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, –5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (–4, 2, –5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (–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 (–3, –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, –4, –7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 – 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 – 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) (–3, 7, 2) and (2, 4, –1)
(iii) (–1, 3, –4) and (1, –3, 4) (iv) (2, –1, 3) and (–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 (–3, 7, 2) and (2, 4, –1)

(iii) Distance between points (–1, 3, –4) and (1, –3, 4)

(iv) Distance between points (2, –1, 3) and (–2, 1, 3)

Class XI Chapter 12 – Introduction to Three Dimensional Geometry Maths

Page 4 of 17

Question 2:
Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
Let points (–2, 3, 5), (1, 2, 3), and (7, 0, –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 – 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, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –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, –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, –2, –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, –5) are positive,
positive, and negative respectively. Therefore, this point lies in octant V.
The x-coordinate, y-coordinate, and z-coordinate of point (–4, 2, –5) are negative,
positive, and negative respectively. Therefore, this point lies in octant VI.
The x-coordinate, y-coordinate, and z-coordinate of point (–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 (–3, –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, –4, –7) are positive,
negative, and negative respectively. Therefore, this point lies in octant VIII.

Class XI Chapter 12 – 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 – 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) (–3, 7, 2) and (2, 4, –1)
(iii) (–1, 3, –4) and (1, –3, 4) (iv) (2, –1, 3) and (–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 (–3, 7, 2) and (2, 4, –1)

(iii) Distance between points (–1, 3, –4) and (1, –3, 4)

(iv) Distance between points (2, –1, 3) and (–2, 1, 3)

Class XI Chapter 12 – Introduction to Three Dimensional Geometry Maths

Page 4 of 17

Question 2:
Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
Let points (–2, 3, 5), (1, 2, 3), and (7, 0, –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 – Introduction to Three Dimensional Geometry Maths

Page 5 of 17

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

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