Chapter 11: Introduction to Three Dimensional Geometry

Summary

• Three-dimensional geometry involves locating points in space using three coordinates.
• Coordinates are represented as (x, y, z) corresponding to distances from three mutually perpendicular planes.
• The three coordinate planes are the XY-plane, YZ-plane, and ZX-plane.
• The origin in three-dimensional space is denoted as O(0, 0, 0).
• The space is divided into eight octants based on the signs of the coordinates.
• The distance between two points in three-dimensional space can be calculated using the formula derived from the Pythagorean theorem.
• Historical contributions to three-dimensional geometry include work by Descartes, Fermat, and Euler.

Introduction to Three Dimensional Geometry

11.1 Introduction

• To locate a point in a plane, two intersecting mutually perpendicular lines (coordinate axes) are needed.
• In three-dimensional space, three numbers (coordinates) are required to represent a point's position.
  • Example: Position of a ball thrown in space or an aeroplane's flight path.
  • Coordinates are the perpendicular distances from three mutually perpendicular planes (e.g., floor and two walls).

11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space

• Three mutually perpendicular planes intersect at a point O, forming:
  • X-axis (horizontal)
  • Y-axis (horizontal)
  • Z-axis (vertical)
• These planes create the XY-plane, YZ-plane, and ZX-plane, dividing space into eight octants.

11.3 Coordinates of a Point in Space

• A point P in space corresponds to an ordered triplet (x, y, z).
• To locate point P:
  1. Drop a perpendicular PM to the XY-plane (M is the foot of the perpendicular).
  2. Draw a perpendicular ML to the x-axis (L).
  3. The coordinates are defined as:
     • OA = x
     • LM = y
     • MP = z
• The coordinates of the origin O are (0, 0, 0).

11.4 Distance between Two Points

• For points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
  • The distance formula is derived from the Pythagorean theorem in three dimensions:
    • PQ² = PA² + AQ²
    • Where PA and AQ are the lengths along the respective axes.

Example Problems

• Example 1: If P is (2, 4, 5), find the coordinates of F.
  • Solution: F's coordinates are (2, 0, 5).
• Example 2: Determine the octant for points (-3, 1, 2) and (-3, 1, -2).
  • Solution: (-3, 1, 2) is in the second octant; (-3, 1, -2) is in octant VI.

Table of Octants

Common Mistakes and Exam Tips in Three Dimensional Geometry

Common Pitfalls

• Misunderstanding Coordinates: Students often confuse the coordinates of points in three-dimensional space. Remember that a point is represented as (x, y, z), where each value corresponds to a distance from the respective coordinate planes.
• Octant Confusion: Identifying the correct octant for a point can be tricky. Ensure you understand the signs of x, y, and z coordinates to determine the octant accurately.
• Distance Formula Errors: When calculating the distance between two points, students may forget to apply the three-dimensional distance formula correctly. The formula is given by PQ² = PA² + AN² + NQ².

Tips for Success

• Visualize the Geometry: Use diagrams to visualize points, lines, and planes in three-dimensional space. This can help in understanding the relationships between different elements.
• Practice with Examples: Work through examples, such as finding the coordinates of points or determining the lengths of segments in three-dimensional figures.
• Review Coordinate Systems: Familiarize yourself with the three coordinate planes (XY, YZ, ZX) and how they divide space into octants. Understanding this will aid in solving problems related to coordinates.
• Check Your Work: Always double-check your calculations, especially when determining distances or coordinates, to avoid simple arithmetic mistakes.