Introduction to Three Dimensional Geometry – Class 11 Mathematics notes, important questions and MCQs

Chapter overview

Points in space have three coordinates (x, y, z). Distance and section formulas extend naturally from two dimensions.

Coordinates

• Three mutually perpendicular axes divide space into 8 octants.
• A point on the x-axis is (x, 0, 0); on the xy-plane (x, y, 0).
• Signs of coordinates tell the octant.

Formulas

• Distance: √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
• Section formula (ratio m : n internally): [(mx₂ + nx₁)/(m + n), …].
• Mid-point: [(x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2]; centroid of a triangle: average of vertices.

Key terms

Octant

one of eight regions formed by the three coordinate planes

Coordinate planes

the xy-, yz- and zx-planes, on which z = 0, x = 0 and y = 0 respectively

Section formula (3D)

point dividing PQ in ratio m : n internally is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n))

Formulas

• d = √(Δx² + Δy² + Δz²)

Important questions with answers

Quick revision

• 8 octants.
• On the yz-plane, x = 0.
• Centroid = average of vertices.

Exam tips

• Distance of (x, y, z) from the origin is √(x² + y² + z²).
• To check collinearity of three points in space, see whether the sum of two distances equals the third.

Practice MCQs

1. Number of octants in space is

   1. 4
   2. 6
   3. 8
   4. 12
   Show answer

   Answer: (c) 8. 2³.

2. A point on the z-axis has the form

   1. (x, 0, 0)
   2. (0, 0, z)
   3. (0, y, 0)
   4. (x, y, 0)
   Show answer

   Answer: (b) (0, 0, z). Only z.

3. Distance between (1, 2, 3) and (4, 6, 3) is

   1. 7
   2. 5
   3. 25
   4. √7
   Show answer

   Answer: (b) 5. √(9 + 16 + 0).

4. The point (−2, 3, −4) lies in the octant with signs

   1. (+, +, +)
   2. (+, −, +)
   3. (−, −, −)
   4. (−, +, −)
   Show answer

   Answer: (d) (−, +, −). Signs of coordinates.

5. The distance of (3, −4, 5) from the x-axis is

   1. √41
   2. 3
   3. 5√2
   4. 4
   Show answer

   Answer: (a) √41. √(y² + z²) = √(16 + 25).

6. Mid-point of (2, 4, 6) and (4, 8, 10) is

   1. (6, 12, 16)
   2. (3, 6, 8)
   3. (1, 2, 2)
   4. (2, 4, 4)
   Show answer

   Answer: (b) (3, 6, 8). Average.

7. Distance of (2, 3, 6) from the origin is

   1. 11
   2. 7
   3. 49
   4. 5
   Show answer

   Answer: (b) 7. √(4 + 9 + 36).

8. The point dividing the join of (1, −2, 3) and (3, 4, −5) in the ratio 2 : 3 internally is

   1. (2, 1, −1)
   2. (9/5, 2/5, −1/5)
   3. (7/5, 0, 1)
   4. (11/5, 2, −3)
   Show answer

   Answer: (b) (9/5, 2/5, −1/5). ((6 + 3)/5, (8 − 6)/5, (−10 + 9)/5).