Need help developing intuition for proving this geometric relationship

Question

I have been studying multivariable calculus, and the course starts with an introduction to vectors. I have been struggling with understanding the mindset needed for proofs with vectors.

As an example, I was working on the problem "Prove using vector methods (without components) that the diagonals of a parallelogram have equal lengths if and only if it is a rectangle."

My immediate thought is that I have to show somehow that the dot product of two vectors originating at a common vertex of the parallelogram is zero. But from here, I write out some relationships between the diagonals and the endpoints, move symbols around the page, until I get something that makes sense.

I have solved this problem already, and for a lot of similar problems I have been working on I could just as easily look at the textbook for an explanation on how to solve it, but that has not been very helpful in helping me understand the mindset or strategy for how to tackle these proofs. Like I said, it feels like I start with some basic idea for what I need to accomplish, and it just turns into my moving symbols around on the page without a clear goal until something workable comes up.

How do I develop the intuition, or perhaps problem-solving mindset to deal with these proofs more effectively? I would also appreciate if someone had resource recommendations for where to read further about learning to prove things

Thanks

Answer 1

The book "How to Prove It: A Structured Approach" by Velleman might be a useful resource.

I personally find the following useful: When reading proofs in textbooks, I try to figure out how the author might have come up with the idea for the proof. Maybe over time this strategy helps to come up with own ideas for proofs.

When writing own proofs, I sometimes try to simplify the statement and first prove a simpler version of it. Then I try whether the same method also works in the more complicated case.

Answer 2

I second Toni’s answer about trying to figure out how the author may have come up with the idea of the proof. Some authors don’t bother with providing any intuition, and the proofs often look “magical” and sometimes rather opaque. Clearly, there are a few arguments for and against this. Such an approach may make things hard to read and unintuitive, but at the same time, maybe that is a challenge to the reader: find out and verify for yourself. George Polya said that math is not a spectator sport. Paul Halmos said “Don’t just read it; fight it! […]”.

This is not to say that intuition shouldn’t be provided; on the contrary, that is usually helpful (whatever works for you — so long as it is accurate!). But sometimes you have to do the hard work to build up some intuition — say by solving many problems, working through the omitted details in a proof, etc. There is something to be said about well-written books with great explanations and intuition, for sure. But no matter how great the book is, I believe that ultimately you still have to solve the problems by yourself if you want to really understand things well and be able to work with them comfortably (unless you are some genius/wizard…).

Speaking of Polya, you may enjoy his book How to Solve It if you haven’t already read it. It can help you with developing that “problem-solving mindset” you inquired about. I also second Toni's recommendation for Velleman's How to Prove It to learn more about proof strategies and proof writing.

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Let's look at your example. I can show how one might approach this by asking a few general questions. Many of these suggestions overlap with what Polya has to say in his book, like asking questions like “What do we want to show?”, “What do we know?”, “Do you know a similar problem?”, etc. (And I am not at all an experienced problem solver — I’m just showing how asking these general questions can make life easier and help to organize things and break the problem into simpler parts.)

What do we want to show? We want to show that the diagonals of a parallelogram have equal lengths if and only if it is a rectangle.

What do we know? (Also: Draw a picture, introduce notation, rephrase things, etc.) I know that we can represent the two sides of the parallelogram by vectors $u$ and $v$. Then the diagonals correspond to the vectors $u + v$ and $u - v$. Therefore the phrase “diagonals of the parallelogram have equal lengths” is equivalent to “$\lVert u + v \rVert = \lVert u - v \rVert$”. Furthermore, if we think about the picture, we see that $u, v$ are perpendicular if and only if we have a rectangle, and I know that $u, v$ being perpendicular means $u \cdot v = 0$. Therefore the phrase “it is a rectangle” is equivalent to “$u \cdot v = 0$”. Combining these two observations, we can now rephrase our question as follows: Show that $\lVert u + v \rVert = \lVert u - v \rVert$ if and only if $u \cdot v = 0$.

In particular, the intuition for our question can in some sense be reduced to the intuition behind why perpendicular means $u \cdot v = 0$, of which there are many great answers that can be easily looked up.

Do you know a similar problem? (Or are there any related facts/knowledge you can use?) Well, maybe I don’t really know a similar problem specifically, but I know that the dot product is the square of the norm. (Maybe I’ve solved many problems with the dot product already, and “moved symbols around” a lot, so I might be familiar with some of its properties.) Specifically, I am reminded of the formula $v \cdot v = \lVert v \rVert^2$.

Now note that the condition “$\lVert u + v \rVert = \lVert u - v \rVert$” means the same as “$\lVert u + v \rVert^2 = \lVert u - v \rVert^2$”, because two nonnegative numbers are equal if and only if their squares are equal. So I have converted this condition into an equality of dot products now! Maybe this gets me closer to trying to show how it is equivalent to the other condition “$u \cdot v = 0$”. So let’s write $\lVert u + v \rVert^2 = \lVert u - v \rVert^2$ as $(u + v) \cdot (u + v) = (u - v) \cdot (u - v)$. I happen to know the basic property of dot product, namely that it is “linear”, so I recognize this as saying that $u \cdot u + 2 (u \cdot v) + v \cdot v = u \cdot u - 2 (u \cdot v) + v \cdot v$. This is equivalent to $2 (u \cdot v) = -2 (u \cdot v)$, i.e. $4 (u \cdot v)$, i.e. $u \cdot v = 0$. The problem is solved.

Polya also recommends looking back at your solution/attempt: once you have written out your answer, examine your solution and check your result. Perhaps you could do some sanity checks. Perhaps you could approach the problem another way or see it with a different perspective. Perhaps there were some important or general strategies you used that could be adopted in future problems. (I can think of a few in this example: (1) Perpendicular means dot product equals zero. (2) If $a, b \geq 0$, then $a = b$ if and only if $a^2 = b^2$. (3) Linearity of the dot product. Etc.)