Cartesian (Euclidean) Plane R2

We choose a coordinate system for the Euclidean plane consisting of two straight lines intersecting at a 90^\circ angle at a point referred to as the origin. One of the lines is called the x_1-axis and the other the x_2-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 (a_1,a_2), where a_1 corresponds to the intersection of the x_1-axis with a line through a parallel to the x_2-axis, and a_2 corresponds to the intersection of the x_2-axis with a line through a parallel to the x_1-axis. The coordinates of the origin are (0,0).

Coordinate system for R^2

In this way, we identify each point a in the plane with its coordinates (a_1,a_2), and we may thus represent the Euclidean plane as R^2, where R^2 is the set of ordered pairs (a_1,a_2) of real numbers a_1 and a_2. That is

  • R^2=\{(a_1,a_2):a_1, a_2\in R\}.

We have already used R^2 as a coordinate system above when plotting/representing a function f:R\rightarrow R, where pairs of real numbers (x,f(x)) are represented as geometrical points in a Euclidean plane on a book-page.

To be more precise, we can identify the Euclidean plane with R^2, 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 x_1-axis is horizontal and increasing to the right, and the x_2-axis is obtained rotating the x_1 axis by 90^\circ, 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 a_2 coordinate positive down, negative up.

Surveyors and Navigators

Recall our friends the Surveyor in charge of dividing land into properties, and the Navigator in charge of steering 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 R^2.

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 (a_1,a_2) 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 now approach to solve using tools of analytic geometry or linear algebra.

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