Vector Proofs

[rebeckab]

Okay, so I think the thing I struggle with in these is actually knowing what I need to prove.. So if I'm asked to prove something is a rhombus, I don't know what I actually need to show that it is a rhombus. Does someone have a list of or know where I can find what I need to show that a rhombus is a rhombus, a kite is a kite yadda yadda?

[paulsterio]

In exams, they usually guide you through the proofs, i.e. part a, part b, part c...etc.

However, in answering your question, well, a rhombus is a shape with a pair of parallel sides and all sides of equal length. So as long as you can prove that is the case, you have a rhombus.

[rebeckab]

Thanks, that's what I've tended to find so far, but I wasn't sure if they would do that in all cases. Should I know them just in case?

[paulsterio]

Well yes, you should know what your basic shapes are from junior mathematics, so what a parallelogram is, what a rhombus is, what a rectangle is, what a square is...etc. You should also be familiar with basic euclidian geometry and all that, basically from Y10 mathematics.

[Lasercookie]

There's these old posts on AN that have a list of this kind of stuff:

[Some] Geometry Prerequisites
Re: Vector proofs intuition.

Those lists aren't really enough on their own though. Like Mr. Stereo said, go refresh your memory on what the definitions for each shape are. Once you've got those in your head it becomes fairly clear what you need to prove and why you're proving that fact.

[rebeckab]

okay.. I know squares and stuff, but I lived overseas for a year on exchange between year 10 and 11, so honestly the only thing I remember from year 10 maths is quadratics and trig. I'll have a look at those posts thouse, thanks

[rebeckab]

Oh those posts are awesome, exactly what I needed. Thanks so much

[BigAl]

check out wikipedia...I know them by heart though. Instead of kite, I preferably use the name 'deltoid' ..it's fancy

[BubbleWrapMan]

Proving that quadrilaterals are specific shapes can be shortened somewhat (as in, you don't have to literally find every vector and length and show that they're equal, or something like that).

If you have a quadrilateral ABCD then proving that (or equivalent) proves that the shape is a parallelogram, and then showing that (or equivalent) will prove it is a rhombus, since a rhombus is a parallelogram.

Then if you wanted to prove, say, ABCD is a square, just prove (or equivalent) since a square is a rhombus which is a parallelogram.

Or for a rectangle, you wouldn't have to prove that that adjacent sides are equal, so you could just do (as a rectangle is a parallelogram) and .

If you consider only parallelograms, rhombuses, squares, and rectangles (which is pretty much all you'd need to know from memory, I doubt you'd need to recall the definition of a kite) then you can see that you'll only need at most three steps to prove a quadrilateral is one of those shapes:

For a parallelogram, you need 1 step (opposite vectors are equal)
For a rhombus, you need 2 steps (prove it is a parallelogram, and then that adjacent sides are equal)
For a rectangle, you need 2 steps (prove it is a parallelogram, and then that adjacent sides are perpendicular)
For a square, you need 3 steps (prove it is a parallelogram, that adjacent sides are equal, and that adjacent sides are perpendicular)

The order doesn't necessarily matter but you should probably start by proving a parallelogram.

A trapezium is a little different, just prove that any two sides are parallel (e.g. for some real k)

For a cyclic quadrilateral you need opposite angles to add up to 180 degrees, so you may not always need to use vectors, but if you do have to use vectors then you can use the fact that if two angles are supplementary then their cosines add to 0 (i.e. ) and use the dot products of appropriate vectors to find the relevant cosines. And you only have to do this with one pair of angles, since it implies that the other pair of angles are also supplementary. (Note: this is assuming you already know the quadrilateral is planar. To prove a quadrilateral is planar you need to prove that any three of the vectors along its edges are linearly dependent, but that probably won't be in an exam.)

I think as far as spesh goes, that's all you need to know about proofs involving quadrilaterals.

[pi]

^You sir, are a boss.

Added a link to that post in the Resources Thread.

[BubbleWrapMan]

Cheers. Hope it helps you guys.