I have always, A.L.W.A.Y.S. found vectors confusing. WHY do they add up in that funny manner. Why do we learn about them at all? They just seem to be a piece of complicated machinery that just makes life difficult for high school and college students. It was only on learning abstract mathematics that I somehow became more comfortable with the need for their existence. I hope high schoolers do not need to go down this long arduous path before finally starting to make sense of things.
Please note that I did not really face a problem with understanding vector properties, or even dealing with vectors in unknown situations. My conundrum was more existential: why did they need to exist at all? Why do they add in that funny way, when numbers don’t? You get the drift. I suppose most people find themselves asking similar questions when they come across vectors.
The key concept that confuses most people here is that of “addition”. We know . ….and so on. The symbol has come to be associated with a very specific situation, in which two real numbers add up to give a third real number (we haven’t crossed over to complex numbers yet in our high school minds).
But then we start finding pointed arrows (vectors) being added to each other. Where did that come from? Clearly vectors are not real numbers! Even though I have finally learned how to add them (make a parallelogram and other arbitrary figures), HOW is this addition?
“Addition” here is just the name of an operator. For real numbers it means something different, and for vectors it means something completely different. This is the result of some very unfortunate nomenclature, fuelling confusion from one generation to another. Hence, when you come across the phrase “vector addition”, do NOT think of addition. Think of the operation your textbook talks about, repeat to yourself “this is not really addition”, and apply the algorithm. You’re set.
The following section is to be read only when you’re comfortable with the process outlined above:
In reality, the nomenclature is not unfortunate. Addition in the real number sense is only a special case of vector addition (which by way of this expression is clearly more general). However, these things become apparent only in retrospect.