# Analytic geometry, proper order for learning and teaching it

1. Jul 6, 2015

### 0kelvin

I'm planning to write about analytical geometry, but I'm unsure about where to start. The course that I've taken begins with the definition of a vector, it then proceeds to develops all the vector algebra and all properties. After it reaches the cross product and triple product it begins to apply all vector algebra that had been developed previously in analytical geometry, ending at conics and quadrics.

However, I've seen courses that begin with linear systems, matrices and matrix algebra, then it introduces vectors. Another approach that I've seen is to begin with coordinate system, lines, points, distance between two points, etc and then it introduces vectors.

I'm not planning to follow the "calculus + analytical geometry" path. Is there a criteria to decide where to start with? The only thing I'm forseeing up to now is that the vector and matrix algebra part is pretty easy to introduce in 2D, then 3D and extend that to $R^{n}$.

Last edited: Jul 6, 2015
2. Jul 6, 2015

### Dr. Courtney

Lots of authors and teachers have differing opinions on the "proper" order for teaching it.

Sometimes, order is influenced by other courses at the school for which a course is a co-requisite.

Instructors in co-requisite courses may request for material to be moved forward so it is learned by their students by the time they need it.

3. Jul 7, 2015

### MidgetDwarf

My book of analytical geometry by Ross R. Middlemiss starts with coordinate system, lines points, distance, then does everything else you mentioned. It is a good book and very small so it fits nicely in a backpack. I can even read it and do problems on public transportation because of its size.

Not sure what level your class it. It sounds like an introduction to linear algebra?

Maybe check out the book by Middlemiss, can be had for 25 cents, five dollars shipped. It is pitched at a high school level, but goes into great detail, as did, much books in those times.

4. Jul 7, 2015

### 0kelvin

Forgot that. The level that I'm going to write is freshman, first year at uni.

The book that I've used follows this order:

- Vector
- Multiplication of vector by a real number
- Adding vector and point
- Application of vectors to plane geometry (barycenter, triangles, trapeze, diagonals, etc) Most teachers postpone this chapter, leaving it to be taught after linear dependance
- Linear dependance
- Basis
- Dot product
- Right hand rule and $V^{3}$
- Cross product
- Triple product
- Coordinate system
- The rest of the book is all analytic gemetry with application of vectors (equations of lines, planes, distance between points, angle between planes, so on. It ends at conics, spheres and quadrics, though most teachers skip this part because the course doesn't have enough time to cover the last part of the book)

Here is the order of a free book I've found:

- Matrices and linear system (covers matrix algebra)
- Inverse matrix and determinants
- Vectors in plane and space (cover all vector algebra)
- Lines and planes
- Conics
- Surfaces and curves in space
- Coordinate system - translation, rotation, identifying conics and quadrics

A third book. Rather tha splitting in analytic geometry and vector algebra and splits in 2D and 3D:

- Coordinates in 2D (distance between point and line, cartesian and other coordinate systems, angle between lines, circumference equation, etc). Fun fac here, this book treats angle between lines before introducing vectors which is the way I was thaugh in high school
- Vectors in 2D (covers vector algebra but operations that require 3D)
- Equations of ellipse, hyperbole and parabola
- Moving from one coordinate system to another
- Linear transformations
- Coordinates in 3D
- From here the book extends the same topics from 2D to 3D (covers cramer's rule, determinants, matrices, calculating areas and volumes, cross product, linear transformations in 3D, completing the squares, etc)

So far I liked the idea of starting with coordinates before vectors. In any case, I'm not going to write a book, but notes online, so I can follow a non linear approach (modular if you will).

Last edited: Jul 7, 2015
5. Jul 7, 2015

### brainpushups

It sounds like you don't have much of a constraint. Have you considered taking the order of topics according to their historical development? You'd probably want to break from the chronology in places seeing as Euclidian vectors were not fully developed in their modern form until near the turn of the 20th century and analytic geometry began with Fermat and Descartes in the 17th century. The third order of topics you listed above appears to be somewhat guided by this approach.

6. Jul 7, 2015

### micromass

Staff Emeritus
The way I would do it would not necessarily be a good way of teaching it.
I would start with Euclidean-like axioms and then use that to define a coordinate system on my plane. Then I would define vectors and prove the vector operations. I would then cover matrices, determinants and their geometric interpretations. I would then go on to conic sections, prove some things using linear algebra. Then I would introduce projective space, and consider the theory of conic sections there. Then I would introduce complex numbers and see what that gives us in geometry. The same thing can be done in 3D space of course.

7. Jul 7, 2015

### 0kelvin

That's the idea. I'll follow that. In addition, I've found lecture notes of a teacher that has this writen in the preface "linear algebra and analytic geometry evolved in parallel, it's impossible to separate the two and as such I'll be using linear algebra, without prior notice, whenever it's applicable".

My idea is somewhat similar to Schaum's Outline, many solved examples.

8. Jul 9, 2015

### mathwonk

50 years ago, analytic geometry meant learning about lines, circles, parabolas, elipses, and hyperbolas (conic sections). there was no mention of linear algebra, matrices, or determinants. those are helpful but they slow down the presentation of the elementary basics which can be taught using trig and algebra. but linear algebra is useful and makes life easier, at least for the author, so it won't hurt you, and may even help you, but if your goal if to cover classical analytic geometry, you are being given an unnecessrily long approach. any old calculus book with a title like "calculus and analytic geometry" will give you the traditional version.

https://www.amazon.com/s/ref=a9_sc_...+and+analytic+geometry&ie=UTF8&qid=1436485477