Chapter 10 Basic Concepts of Descriptive Vector Geometry

Section 10.1 From Arrows to Vectors

10.1.2 Coordinate Systems in Three-Dimensional Space

In Section 9.1 of the previous Module 9 we introduced Cartesian coordinate systems and points in the plane described by coordinates with respect to these coordinate systems. A solid understanding of these concepts is now presumed in this Module. To describe a point in three-dimensional space, three coordinates are required. Thus, a coordinate system in three-dimensional space needs three axes, the x-axis, y-axis and z-axis (sometimes called the x1 -axis, x2 -axis, and x3 -axis). Usually, points will be denoted by upper-case Latin letters P,Q,R,, and their coordinates will be denoted by lower-case Latin letters a,b,c,x,y,z, as variables. Extending the notation from Module 9, the coordinates of a point are written, for example, as follows:




Here, the x-coordinate of the point Q is -2, its y-coordinate is 2 and its z-coordinate is  3. The point with the coordinates (0;0;0) is called the origin, and it is denoted by the symbol O. All these points are drawn in the figure below.
The dashed lines in this figure indicate how the coordinates of points in such a three-dimensional representation can be drawn and read off. Note that these lines are all parallel to the coordinate axes.
We will only consider coordinate systems in three-dimensional space with perpendicular coordinate axes - these are Cartesian coordinate systems. Furthermore, we will use the common mathematical convention that coordinate systems in three-dimensional space are right-handed. Sometimes these are also called positively oriented. This means that the positive directions of the x, y, and z-axis can be determined by means of the right-hand rule as illustrated in the figure below.

However, there are various possible representations. In the figure above showing the points P and Q, the x-axis points to the right, the y-axis points up, and the z-axis points perpendicularly outwards from the drawing plane. In the figure which illustrates the right-hand rule, the x-axis points to the right, the y-axis points backwards into the drawing plane, and the z-axis points up. However, both coordinate systems are right-handed.
Exercise 10.1.1
Specify the coordinates of the points indicated in the figure below. Consider how all indicated points can be collected into one mathematical object.
  1. A = .
  2. B = .
  3. C = .
  4. D = .
  5. P = .
Enter points in the form (x;y;z). Enter, for example, (8;-4;15) for the point with x-coordinate 8, y-coordinate -4, and z-coordinate 15.

As in the two-dimensional case discussed in Section 9.1.2, an arbitrary number of points in three-dimensional space can be collected into a set of points. The following notation is used:
Info 10.1.2
The set of all points (in space) specified as coordinate triples with respect to a given Cartesian coordinate system is denoted by

3 :={(x;y;z):xyz}.

The symbol 3 reads as " three" or " to the power of three". This indicates that a point can be uniquely described by a coordinate triple (also known as an ordered triple) consisting of three real numbers.