Elementary analytic geometry textbook recommendation

[LittleRookie]

Every explanation about scaling a 2D vector, or equivalently having a line segment PQ on cartesian plane and then find a point R on the line PQ satisfying PR/PQ = r (fixed given r) starts with that one specific case in the picture. A formula for the coordinates of R is then given for that case.

However, that is also the only case that is covered. The cases whereby the slope is non-positive and the line segment PQ is vertical are not shown to share the same formula. Also, the case of point Q being the head of the vector is not proved.

Furthermore, they will then tell you that the case for r>1 is equivalent to R extending from the segment PQ and the case r is negative is simply all the above but now put R in the opposite direction, and "miraculously" the same formula given at the start will work.

And lastly, they don't even mention that suppose a point on the cartesian plane satisfy the formula, then they are the point R as discussed in the theorem. i.e. the converse of the theorem.

It's so frustrating to need to spot incomplete proofs and filling up the proofs by myself when the book is supposed to teach me. Does anyone know of an elementary analytic geometry book with complete proofs?

 

[BvU]

the book is supposed to teach me

That's not exactly how math works. You are supp0sed to teach yourself by understanding the subject matter, follow through some examples and do the exercises (that often bring up questions like yours).
The book you are looking for would be too thick to handle, e.g. like

 

[LittleRookie]

That's not exactly how math works. You are supp0sed to teach yourself by understanding the subject matter, follow through some examples and do the exercises (that often bring up questions like yours).
The book you are looking for would be too think to handle, e.g. like

I meant if the author discusses a theorem and its proof, then the least that the author can do is to provide a complete proof, or if the proof requires further knowledge, the author can mention which cases are not provided with proof because so and so. Furthermore, the complete proof of the theorem that I've mentioned does not require further knowledge.

I agree that we have to understand the proof on our own (is that what you meant by teach yourself?) However, you can't if the complete proof is not provided. The student will then end up looking for a complete proof elsewhere.

 

[Mark44]

View attachment 254672Every explanation about scaling a 2D vector, or equivalently having a line segment PQ on cartesian plane and then find a point R on the line PQ satisfying PR/PQ = r (fixed given r) starts with that one specific case in the picture. A formula for the coordinates of R is then given for that case.

However, that is also the only case that is covered. The cases whereby the slope is non-positive and the line segment PQ is vertical are not shown to share the same formula.

It doesn't make any difference if the slope of the segment PQ is negative. The ratio PR/PQ will still be a positive number as long as R lies between P and Q.

Also, the case of point Q being the head of the vector is not proved.

For the vector $\overline{PQ}$, Q is the head of the vector. Did you mean when P is the head of the vector? I.e., $\overline{QP}$.

Furthermore, they will then tell you that the case for r>1 is equivalent to R extending from the segment PQ and the case r is negative is simply all the above but now put R in the opposite direction, and "miraculously" the same formula given at the start will work.

And lastly, they don't even mention that suppose a point on the cartesian plane satisfy the formula, then they are the point R as discussed in the theorem. i.e. the converse of the theorem.

It's so frustrating to need to spot incomplete proofs and filling up the proofs by myself when the book is supposed to teach me. Does anyone know of an elementary analytic geometry book with complete proofs?

As @BvU already mentioned, no textbooks will provide complete proofs of every possible variation of every statement. Many textbooks will leave things with "the proof is left to the reader."

 

[MidgetDwarf]

Although not sufficient. Have you looked into something called the Ruler Postulate and to a greater extant, the definition of a metric? As someone mentioned above, we could define the length of line segments as ratios. In one representation of Hyperbolic Geometry, we use something called a cross ratio.

 

[MidgetDwarf]

I meant if the author discusses a theorem and its proof, then the least that the author can do is to provide a complete proof, or if the proof requires further knowledge, the author can mention which cases are not provided with proof because so and so. Furthermore, the complete proof of the theorem that I've mentioned does not require further knowledge.

I agree that we have to understand the proof on our own (is that what you meant by teach yourself?) However, you can't if the complete proof is not provided. The student will then end up looking for a complete proof elsewhere.

I think you are approaching the study of mathematics the wrong way...

 

[LittleRookie]

I think you are approaching the study of mathematics the wrong way...

Why so? Please enlighten me.

 

[Mark44]

Every explanation about scaling a 2D vector, or equivalently having a line segment PQ on cartesian plane and then find a point R on the line PQ satisfying PR/PQ = r (fixed given r) starts with that one specific case in the picture. A formula for the coordinates of R is then given for that case.

However, that is also the only case that is covered. The cases whereby the slope is non-positive and the line segment PQ is vertical are not shown to share the same formula. Also, the case of point Q being the head of the vector is not proved.

I think you are approaching the study of mathematics the wrong way...

Why so? Please enlighten me.

@LittleRookie, the comment by @MidgetDwarf might relate to your opening post in this thread. In your post you mention an explanation about scaling a 2D vector and complain that every possible case is not proved. Although the image you copied shows a vector in the first quadrant that has a positive slope, the points at either end of the vector are arbitrary, so don't depend on any particular orientation of the vectors involved. I think I already mentioned that it makes no difference about the slope being non-positive. Further, it makes no difference that I can see if the vectors are vertical, or even if the vectors point in opposite directions.

It seems to me that you are saying that an explanation of a calculation must show the same rigor as the proof of a theorem, which is not necessarily the case. For your example, and the cases you say are being omitted, it should be a simple matter to fill in the steps.