Question

The focus of this unit is understanding geometry using positions of points in a Cartesian coordinate system. The study of the relationship between algebra and geometry was pioneered by the French mathematician and philosopher René Descartes. In fact, the Cartesian coordinate system is named after him. The study of geometry that uses coordinates in this manner is called analytical geometry.
How does this approach differ from other approaches to geometry that you have used or seen? Based on your experiences so far, which approach to geometry do you prefer? Why? Which approach is easier to extend beyond two dimensions? What are some situations in which one approach to geometry would prove more beneficial than the other? Describe the situation and why you think one approach is more applicable

Answer 1

1. How does this approach differ from other approaches to geometry that you have used or seen?

From the point of view of analytical geometry you need to use coordinate systems, this contrasts with synthetic geometry that is a type of geometry without using the coordinate systems or formulas. For example, consider a line segment that is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its endpoints. This is a simple concept of elemental geometry, but if you consider this segment as a targeted segment, then this is a case of analytical geometry. Therefore, in this type of geometry you need to consider the direction of the segment. On the other hand,  regarding the length of the segment, in elemental geometry, AB = BA, but in analytical geometry AB = -BA.

2. Based on your experiences so far, which approach to geometry do you prefer?

I prefer analytical geometry rather than the others.

3. Why?

Because it is applicable in many subjects. For example, electromagnetic theory, antennas, microwaves, among other. It would be impossible the study of these subjects using only elemental geometry or others. So, you need to find mathematical models which are found only using analytical geometry.

4. Which approach is easier to extend beyond two dimensions? 

Other types of geometry are easier. For example, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space.
For instance, stereometry deals with the measurements of volumes of various solid figures, that are three-dimensional figures. This includes pyramids, prisms, and other polyhedrons; cylinders; cones; truncated cones, and balls bounded by spheres. So, the approaching to this subject using analytical geometry is more complex because you must consider coordinates systems.

5. What are some situations in which one approach to geometry would prove more beneficial than the other?

In the field of electromagnetic theory, analytical geometry is very useful. You find mathematical models that are very important in this field. According to the geometry of any problem you need to use an specific coordinate system. There are three fundamental systems used in this field, namely: Cartesian, cylindrical and spherical coordinate systems. From then on, you will find equations that adjust to the problems used in many applications.

6. Describe the situation and why you think one approach is more applicable

In telecommunications, you model antennas using coordinates systems. You set an antenna you are studying in a coordinate system. From then on, you find equations that adjust to the real model and can predict things according to that mathematical model. Besides, the mathematical model that results from the study of the electromagnetic waves is applied using this geometry making possible the communications via telephone.