Vector proofs intuition.

[rife168]

I find vector proofs quite annoying in the fact that I understand quite well how many different proofs work, but I struggle a little bit about constructing my own, and then knowing if it is actually valid.
So, how do you go about vector proofs?

[TrueTears]

one of the biggest areas of vector proofs (and proofs in general) is knowing what you have to prove, in other words, what mathematical definitions do you have to prove?

One of the areas in vector proofs i think in vce revolves around geometric shapes, rectangles, squares and all that shizz, one really important thing is to know what are the mathematical definitions of a square, rhombus etc then prove the mathematical definition.

Most of the confusion comes from the fact that you know what you sort of have to prove (intuitively) but don't know exactly what you have to prove (rigorously) to get the final result, the best way of improving this is to just simply know your definitions, do different types of questions and remember the crux of each question.

[Special At Specialist]

TrueTears is right. For example, if you wanted to prove a square, you have to prove that all four corners are perpendicular and that all lengths are the same.

I struggle with vector proofs too, though. I got up to a question in the textbook which asked me to "Prove the cosine rule for any triangle". After about an hour, I finally got an answer, but then got so frustrated that I skipped the rest of that exercise.

[kamil9876]

After about an hour, I finally got an answer, but then got so frustrated that I skipped the rest of that exercise.

There's nothing wrong with that, I think solving a problem independantly after one hour is worth more than someone giving u the answer in one minute. When doing it for the first time such an approach is ok, eventually you will get good at it. I think people are often scared of vector proofs because they can't do it after 1minute of thinking, and then they panic because on an exam you only have a limited time, but you can always learn first then prepare for exam later.

Some mathematical advice: draw a picture to see why it's true, try to convert picture into formal things like vectors, dot products(you will get better at this with experience) etc. If you want to work with instead work with because dot products have nice properties, such as , this leads to (quite useful from experience).

[VCE_2012]

Before you start doing vector proofs  it is useful to know some fundamental vector proofs* and understand the basic rules such as the dot product etc. You can also learn from other text book examples (try to source your knowledge not just from your school's text book, this applies to all topics in SPM).

By the way you may encounter 'crazy proofs' in some text books, don't despair when you see them because they won't probably appear on your exam because the study design states 'vector proofs of simple geometric results'. 

* Vector proof: of the cosine rule, Pythagorean theorem, diagonals of a parallelogram bisect etc









* ( such as the 'cosine proof', 'Pythagoras theorem', how to prove a 'square' etc)

[paulsterio]

Vector proofs in Exams aren't usually tough. They're mostly graduated, so they will have part a), part b), part c) and then at the end it might be something like "Hence prove that ABCD is a parallelogram". So it's a lot easier than textbook questions.

Also, a lot of the time, the answers to the earlier parts are given, so it might be a "Show that a = 5 and b = 3" type question, which means if you don't get part a), you'll still be able to go ahead.

[yawho]

In the study design an example: vector proof of the medians of a triangle are concurrent. How do you prove it?

[VCE_2012]

In the study design an example: vector proof of the medians of a triangle are concurrent. How do you prove it?

That is a proof worth knowing. It is in the examples of the vector proof chapter in the essentials text book.

[paulsterio]

That's a really tough proof, the fact that the medians of a triangle are co-current.

Basically the triangle will have 3 medians. You'll have to find the point of intersection between two of them and then the other two and then prove that they are the same point.

[Special At Specialist]

That's a really tough proof, the fact that the medians of a triangle are co-current.

Basically the triangle will have 3 medians. You'll have to find the point of intersection between two of them and then the other two and then prove that they are the same point.

If we assume that there is already a point, X, where the lines from the vertices to the opposite median side lengths intersect, then I can prove that they intersect at a point which is 2/3rds of the way from the vertex to the median side length:
http://img99.imageshack.us/img99/1610/mathsproblem14.png
http://img138.imageshack.us/img138/2744/mathsproblem15.png
But how do I prove that they intersect in the first place? Is what I have done by itself sufficient?

[MichaelT]

Hmm ye vector proofs tend to be quite confusing, just a new concept which makes it confusing.

[Sam_95]

I've been trying to do some of these Vector proofs, and I just don't get them for the life of me. We have been doing them from the 'Essential Book.'

Our Spesh teacher told us the ones that come in the exams are no where near as hard as the ones in the book. Is this true?

[Ravit]

Yes i have heard the exam questions being quite bit easier than those in the essentials book.
They would be, because some of the questions in the exercise require a significant amount of time, something you won't have very much of in the exam.

[Bhootnike]

Yes i have heard the exam questions being quite bit easier than those in the essentials book.
They would be, because some of the questions in the exercise require a significant amount of time, something you won't have very much of in the exam.

just imagine the assessors going through 5000 exams or whatever it is, to some complex vector proof question, to which students have given 1000000 different methods!
i think its for that reason, the exam questions are straightforward and simple..

My own question,
can anyone link or .. tell haha, what the geometric definitions of 'vce accredited' (lol) shapes? id wanna rote learn , what i wanna show for each question so it becomes natural. so e.g., parallelogram OABC,     

O       A
C       B

if asked to show that it is a parallelogram, you'd would wanna show that  OC=AB, AND OA = CB  yeah? hopefully thatss right haha

but yeah, ,could someone hit us/me up with a summary of what youd have to show, for prooving the shapes tested at vce level.. ?! pleaseeeeeeeeeee

[TrueTears]

Square is a special case of 2 shapes, Rectangle (equal angles) or Rhombus (equal sides).
Rectangle and Rhombus are both special cases of a parallelogram (2 parallel sides). The parallelogram is a special case of the trapezium (1 parallel side). The trapezium and the cyclic quadrilateral are special cases of a quadrilateral.

For a square you need to prove
1. Angles are equal
2. Opposite sides are congruent.

For a rectangle: All angles are right angles.

For a rhombus: Opposite sides are congruent.

For a parallelogram: Opposite sides are parallel.

For a trapezium: 1 pair of opposite sides are parallel.

For a cyclic quadrilateral: Opposite angles are supplementary