Class 11 Mathematics Notes Chapter 12 (Introduction to three dimensional geometry) – Mathematics Book

Alright class, let's begin our study of three-dimensional geometry. This chapter extends the concepts you learned in two-dimensional coordinate geometry to space. Understanding 3D space is crucial not just for mathematics, but also for various fields like physics, engineering, computer graphics, and more. For your government exam preparation, mastering the fundamentals – coordinates, distance, and division of segments – is key.

Chapter 12: Introduction to Three Dimensional Geometry - Detailed Notes

1. Introduction: From 2D Plane to 3D Space

• In two-dimensional (2D) geometry, we locate a point in a plane using two perpendicular axes (X and Y) and an ordered pair of coordinates (x, y).
• In three-dimensional (3D) geometry, we locate a point in space using three mutually perpendicular axes (X, Y, and Z).

2. Coordinate Axes and Coordinate Planes in 3D Space

• Coordinate Axes: Three mutually perpendicular lines intersecting at a point called the Origin (O).
  • X-axis, Y-axis, Z-axis.
  • Conventionally, they form a right-handed system. If you curl the fingers of your right hand from the positive X-axis towards the positive Y-axis, your thumb points along the positive Z-axis.
• Origin (O): The point of intersection of the three axes. Its coordinates are (0, 0, 0).
• Coordinate Planes: The three planes determined by pairs of axes.
  • XY-plane: The plane containing the X and Y axes. The equation of this plane is z = 0.
  • YZ-plane: The plane containing the Y and Z axes. The equation of this plane is x = 0.
  • XZ-plane: The plane containing the X and Z axes. The equation of this plane is y = 0.
• These three planes divide the space into eight regions called Octants.

3. Coordinates of a Point in Space

• Any point P in space is represented by an ordered triplet of real numbers (x, y, z).
• x-coordinate: The perpendicular distance from point P to the YZ-plane.
• y-coordinate: The perpendicular distance from point P to the XZ-plane.
• z-coordinate: The perpendicular distance from point P to the XY-plane.

4. Octants and Signs of Coordinates

The signs of x, y, and z determine the octant in which a point lies.

• Important Note: Points on the coordinate planes or axes do not lie in an octant.
  • Point on XY-plane: (x, y, 0)
  • Point on YZ-plane: (0, y, z)
  • Point on XZ-plane: (x, 0, z)
  • Point on X-axis: (x, 0, 0)
  • Point on Y-axis: (0, y, 0)
  • Point on Z-axis: (0, 0, z)

5. Distance between Two Points

• Let P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) be two points in space.
• The distance between P and Q, denoted by PQ, is given by the distance formula (derived using Pythagoras theorem twice):
  PQ = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
• Distance from Origin: The distance of a point P(x, y, z) from the origin O(0, 0, 0) is:
  OP = √[(x - 0)² + (y - 0)² + (z - 0)²] = √[x² + y² + z²]

Example: Find the distance between A(1, -2, 3) and B(4, 2, -1).
AB = √[(4 - 1)² + (2 - (-2))² + (-1 - 3)²]
AB = √[(3)² + (4)² + (-4)²]
AB = √[9 + 16 + 16] = √41 units.

6. Section Formula

This formula helps find the coordinates of a point that divides the line segment joining two given points in a specific ratio.

• Let R(x, y, z) be a point that divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in the ratio m:n.

  • (a) Internal Division: If R divides PQ internally (R lies between P and Q).
    x = (mx₂ + nx₁)/(m + n)
    y = (my₂ + ny₁)/(m + n)
    z = (mz₂ + nz₁)/(m + n)

  • (b) External Division: If R divides PQ externally (R lies on the line extended beyond P or Q).
    x = (mx₂ - nx₁)/(m - n)
    y = (my₂ - ny₁)/(m - n)
    z = (mz₂ - nz₁)/(m - n)
    (Note: Often derived by replacing 'n' with '-n' in the internal division formula).

  • (c) Mid-Point Formula: If R is the midpoint of PQ, then m = n = 1 (ratio 1:1). The formula simplifies to:
    x = (x₁ + x₂)/2
    y = (y₁ + y₂)/2
    z = (z₁ + z₂)/2

Example: Find the coordinates of the point which divides the line segment joining A(-2, 3, 5) and B(1, -4, 6) internally in the ratio 2:3.
Here, (x₁, y₁, z₁) = (-2, 3, 5), (x₂, y₂, z₂) = (1, -4, 6), m = 2, n = 3.
x = (21 + 3(-2))/(2 + 3) = (2 - 6)/5 = -4/5
y = (2*(-4) + 33)/(2 + 3) = (-8 + 9)/5 = 1/5
z = (26 + 3*5)/(2 + 3) = (12 + 15)/5 = 27/5
The coordinates are (-4/5, 1/5, 27/5).

7. Centroid of a Triangle

• If A(x₁, y₁, z₁), B(x₂, y₂, z₂), and C(x₃, y₃, z₃) are the vertices of a triangle, then the coordinates of its centroid G are given by:
  G = ( (x₁ + x₂ + x₃)/3 , (y₁ + y₂ + y₃)/3 , (z₁ + z₂ + z₃)/3 )
• (The centroid is the point of intersection of the medians and divides each median in the ratio 2:1).

Key Formulas Summary:

• Distance: PQ = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
• Distance from Origin: OP = √[x² + y² + z²]
• Internal Section: R = ( (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n) )
• Mid-Point: M = ( (x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2 )
• Centroid: G = ( (x₁ + x₂ + x₃)/3 , (y₁ + y₂ + y₃)/3 , (z₁ + z₂ + z₃)/3 )

Remember to practice problems applying these formulas. Identifying the correct formula and substituting values carefully is crucial for competitive exams.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed:

1. The point (-3, 4, -5) lies in which octant?
   (A) II
   (B) III
   (C) VI
   (D) VII

2. What are the coordinates of any point on the YZ-plane?
   (A) (x, y, 0)
   (B) (0, y, z)
   (C) (x, 0, z)
   (D) (0, 0, z)

3. The distance of the point P(3, -4, 5) from the origin is:
   (A) 5
   (B) 5√2
   (C) 50
   (D) √50

4. Find the distance between the points A(2, 3, 5) and B(4, 3, 1).
   (A) √12
   (B) √20
   (C) √24
   (D) 6

5. The coordinates of the midpoint of the line segment joining the points (2, -1, 5) and (4, 3, -1) are:
   (A) (3, 1, 2)
   (B) (6, 2, 4)
   (C) (1, 2, -3)
   (D) (3, -2, 3)

6. If a point lies on the Z-axis, what are its x and y coordinates?
   (A) x=0, y=1
   (B) x=1, y=1
   (C) x=1, y=0
   (D) x=0, y=0

7. The point (5, -2, 0) lies on which coordinate plane?
   (A) XY-plane
   (B) YZ-plane
   (C) XZ-plane
   (D) None of these

8. Find the coordinates of the point which divides the line segment joining (-1, 7, 2) and (3, -1, 10) internally in the ratio 1:3.
   (A) (0, 5, 4)
   (B) (1, 3, 6)
   (C) (2, -1, 8)
   (D) (0, 5, 6)

9. The three coordinate planes divide the space into how many octants?
   (A) 3
   (B) 4
   (C) 6
   (D) 8

10. The vertices of a triangle are A(1, 0, 2), B(3, 1, 4), and C(5, 0, 6). The centroid of the triangle is:
    (A) (3, 1/3, 4)
    (B) (9, 1, 12)
    (C) (3, 1, 4)
    (D) (4, 1/3, 3)

Answer Key:

1. (D) VII (x=-ve, y=-ve, z=-ve is VII, but x=-ve, y=+ve, z=-ve is VI. Let's recheck signs: (-3, 4, -5) -> x is negative, y is positive, z is negative. This corresponds to Octant VI.) Correction: Let's re-verify the octant signs. I: (+,+,+), II: (-,+,+), III: (-,-,+), IV: (+,-,+), V: (+,+, -), VI: (-,+, -), VII: (-,-, -), VIII: (+,-, -). So, (-3, 4, -5) has signs (-, +, -), which is Octant VI. My initial answer D was incorrect. The correct answer is (C).
2. (B) (0, y, z)
3. (D) √50 (√(3² + (-4)² + 5²) = √(9 + 16 + 25) = √50)
4. (B) √20 (√[(4-2)² + (3-3)² + (1-5)²] = √[2² + 0² + (-4)²] = √[4 + 0 + 16] = √20)
5. (A) (3, 1, 2) ( [(2+4)/2, (-1+3)/2, (5-1)/2] = [6/2, 2/2, 4/2] = (3, 1, 2) )
6. (D) x=0, y=0
7. (A) XY-plane (z-coordinate is 0)
8. (A) (0, 5, 4) ( x = [13 + 3(-1)]/(1+3) = (3-3)/4 = 0; y = [1*(-1) + 37]/(1+3) = (-1+21)/4 = 20/4 = 5; z = [110 + 3*2]/(1+3) = (10+6)/4 = 16/4 = 4 )
9. (D) 8
10. (A) (3, 1/3, 4) ( [(1+3+5)/3, (0+1+0)/3, (2+4+6)/3] = [9/3, 1/3, 12/3] = (3, 1/3, 4) )

Corrected Answer Key:

1. (C)
2. (B)
3. (D)
4. (B)
5. (A)
6. (D)
7. (A)
8. (A)
9. (D)
10. (A)

Study these concepts and formulas thoroughly. Practice applying them to different types of problems. Good luck with your preparation!