4. Co-ordinate Geometry

Fig. 4.1 René Descartes (1596 - 1650)

The area of mathematics dealing with describing geometry in terms of coordinate systems, hence as points and vectors, is called coordinate geometry or Cartesian geometry (named after French mathematician and philosopher René Descartes). In this chapter we systematise the basic knowledge and understanding of points, lines and planes.

4.1. Points

In his works the Elements, Euclid described a point as “that which has no part”. A point has position only, it has no length, width or thickness, and thus no area or volume. As we have already seen, a typical point in \(\mathbb{R}^n\) is described by its co-ordinates \((a_1,a_2,\dots,a_n)\), where the \(a_i \in \mathbb{R}\). We also think of \(\mathbb{R}^0\) as a single point. This might seem a bit strange because what does a \(0\)-tuple even mean? However, it makes sense for us to adopt this absolutely standard convention.

Fig. 4.2 The position of a point in \(\mathbb{R}^3\) can be defined by its co-ordinates \((x, y, z)\).