#### 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 ${x}_{1}$-axis, ${x}_{2}$-axis, and ${x}_{3}$-axis). Usually, points will be denoted by upper-case Latin letters $P,Q,R,\dots$, and their coordinates will be denoted by lower-case Latin letters $a,b,c,x,y,z,\dots$ as variables. Extending the notation from Module 9, the coordinates of a point are written, for example, as follows:

$P=\left(1;3;0\right)$

or

$Q=\left(-2;2;3\right) .$

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 $\left(0;0;0\right)$ 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 $\left(x;y;z\right)$. 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

$ℝ{}^{3}:=\left\{\left(x;y;z\right) : x\in ℝ\wedge y\in ℝ\wedge z\in ℝ\right\} .$

The symbol $ℝ{}^{3}$ reads as "$ℝ$ three" or "$ℝ$ 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.