Cartesian (Euclidean) Plane R2

We choose a coordinate system for the Euclidean plane consisting of two straight lines intersecting at a 90 degrees angle at a point referred to as the origin. One of the lines is called the x1-axis and the other the x2-axis, and each line is a copy of the real line R. The coordinates of a given point a in the plane is the ordered pair of real numbers (a1,a2), where a1 corresponds to the intersection of the x1-axis with a line through the point a parallel to the x2-axis, and a2 corresponds to the intersection of the x2-axis with a line through a parallel to the x1-axis. The coordinates of the origin are (0,0).

Coordinate system for R2.

In this way, we identify each point a in the plane with its coordinates (a1,a2) and we may thus represent the Euclidean plane as R2, where R2 is the set of ordered pairs (a1,a2) of real numbers a1 and a2. In other words with set notation {}

  • R2={(a1,a2): a1, a2 ∈ R}.

We have already used R2 as a coordinate system above when plotting/representing a function f(x) of a real variable x, where pairs of real numbers  (x,f(x)) are represented as geometrical points in a Euclidean plane on a book-page or screen.

To be more precise, we can identify the Euclidean plane with R2 once we have chosen the (i) origin, and the (ii) direction and  (iii) scaling of the coordinate axes.

There are many possible coordinate systems with different origins and orientations/scalingsof the coordinate axes, and the coordinates of a geometrical point depend on the choice of coordinate system. The need to change coordinates from one system to another thus quickly arises, and will be an important topic below.

Often, we orient the axes so that the x1-axis is horizontal and increasing to the right, and the x2-axis is obtained rotating the  x1 axis by 90 degrees or a quarter of a complete revolution counter-clockwise, with the positive direction of each coordinate axis may be indicated by an arrow in the direction of increasing coordinates.

However, this is just one possibility. For example, to describe the position of points on a computer screen or a window on such a screen, it is not uncommon to use coordinate systems with the origin at the upper left corner and counting the a2 coordinate positive down, negative up.

Surveyors and Navigators

The task of a Surveyor can be to divide land into properties, and that of a Navigator to steer a ship. In both cases we assume that the distances involved are sufficiently small tomake the curvature of the Earth negligible, so that we may view the world as R2.

Basic problems faced by a Surveyor are (s1) to locate points in Nature with given coordinates on a map and (s2) to compute the area of aproperty knowing its corners.

Basic problems of a Navigator are (n1) to find the coordinates on a map of his present position in Nature and (n2) to determine the present direction to follow to reach a point of destiny.

We know from Chapter 2 that problem (n1) may be solved using a GPS navigator, which gives the coordinates (a1,a2) of the current position of the GPS-navigator at a press of a button. Also problem (s1) may be solved using a GPS-navigator iteratively in an `inverse” manner: press the button and check where we are and move appropriately if our coordinates are not the desired ones. In practice, the precision of the GPS-system determines its usefulness and increasing the precision normally opens a new area of application. The standard GPS with a precision of 10 meters may be OK for a navigator, but not for a surveyor, who would liketo get down to meters or centimeters depending on the scale of the property.

Scientists measuring continental drift or beginning landslides, use an advanced form of GPS with a precision of millimeters.

Having solved the problems (s1) and (n1) of finding the coordinates of a given point in Nature or vice versa, there are many related problems of type (s2) or (n2) that can be solved using mathematics, such as computing the area of pieces of land with given coordinates or computing the direction of a piece of a straight line with given start and end points.

These are examples of basic problems of geometry which we approach using tools of analytic geometry or linear algebra.

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