Introduction to Three Dimensional Geometry

11.1 Introduction

In this introductory section, the concept of extending coordinate geometry from two dimensions to three dimensions is established. In two-dimensional geometry, the position of a point in a plane is located using two mutually perpendicular lines called coordinate axes, and the point is represented by two numbers called coordinates. However, in real life, many objects and points exist in three-dimensional space rather than just on a plane. For example, the position of a ball thrown in space at different times or the position of an airplane flying from one place to another cannot be described using only two coordinates. Similarly, to locate the position of the lowest tip of an electric bulb hanging from the ceiling or the central tip of a ceiling fan in a room, we need not only the perpendicular distances from two adjacent walls but also the height from the floor. This necessitates the use of three numbers representing the perpendicular distances from three mutually perpendicular planes: the floor and two adjacent walls. These three numbers are called the coordinates of the point with respect to the three coordinate planes. Thus, a point in space is represented by three coordinates. This chapter introduces the basic concepts of geometry in three-dimensional space, laying the foundation for further study of spatial geometry and analytical geometry in three dimensions.

• Two-dimensional coordinate geometry uses two perpendicular axes to locate points in a plane.
• Real-life objects often exist in three-dimensional space requiring three coordinates.
• Three mutually perpendicular planes (floor and two walls) define the coordinate system in space.
• Coordinates in three-dimensional space are three numbers representing perpendicular distances from these planes.
• This chapter focuses on the fundamentals of three-dimensional geometry.
• Understanding three-dimensional coordinates is essential for describing spatial positions.
• 📌 Coordinate axes: Two mutually perpendicular lines used to locate points in a plane.
• 📌 Coordinates: Numbers representing the position of a point with respect to coordinate axes or planes.
• 📌 Three-dimensional space: Space defined by three mutually perpendicular directions or planes.

11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space

This section introduces the three-dimensional rectangular coordinate system. Consider three planes intersecting at a point O such that these planes are mutually perpendicular to each other. These three planes intersect along three lines: X'OX, Y'OY, and Z'OZ, which are called the x-axis, y-axis, and z-axis, respectively. These axes are mutually perpendicular and form the rectangular coordinate system in space. The planes formed by pairs of these axes are called coordinate planes: the XOY plane (XY-plane), the YOZ plane (YZ-plane), and the ZOX plane (ZX-plane). The XOY plane is taken as the plane of the paper, and the Z'OZ line is perpendicular to this plane, considered vertical if the paper is horizontal. Distances measured upward from the XY-plane along OZ are positive, and those downward along OZ' are negative. Similarly, distances to the right of the ZX-plane along OY are positive, and to the left along OY' are negative. Distances in front of the YZ-plane along OX are positive, and behind along OX' are negative. The origin O is the point of intersection of the three axes. The three coordinate planes divide the space into eight parts called octants, labeled I to VIII, with each octant corresponding to different sign combinations of the coordinates (x, y, z). This coordinate system forms the basis for locating points in three-dimensional space.

• Three mutually perpendicular planes intersect at origin O forming the 3D coordinate system.
• Axes X'OX, Y'OY, Z'OZ are mutually perpendicular lines called x, y, and z axes.
• Coordinate planes are XY (XOY), YZ (YOZ), and ZX (ZOX) planes formed by pairs of axes.
• Positive and negative directions along axes are defined relative to coordinate planes.
• The three coordinate planes divide space into eight octants with distinct sign patterns.
• Origin O is the intersection point of all three axes.
• 📌 Coordinate axes: x, y, z axes intersecting at origin and mutually perpendicular.
• 📌 Coordinate planes: XY, YZ, ZX planes formed by pairs of coordinate axes.
• 📌 Octants: Eight parts of space divided by the three coordinate planes.