Are the definitions of dot product and cross product the wrong way round? This is something that has been bugging me since I started studying again. It seems to me that the definitions and symbols for the dot and cross products are backwards.
When I first learnt multiplication at school it was with the symbol 'x' as in times tables and all that. This is the symbol I'm most familiar with for multiplication as is the majority of people. When I was in the last years of high school studying mechanics I was introduced to dot products and cross products. The dot product is the same as the multiplication I have always done but has a different symbol. The cross product is a new special type of multiplication that uses the 'x' symbol I was used to for multiplying numbers.
I am used to using the 'x' to multiplying numbers - it would confuse me and most people that looked at my notes if I started putting the dot symbol in instead, so I would rather not.
Why isn't the operation known as the dot product denoted with the classic 'x' symbol and the new symbol used for the thing known as the cross product.
I have studied maths to high school level - carrying on till I was 18. I don't see myself studying to a higher level - I have almost reached the level of maths I can understand. I would appreciate if someone could explain to me if I am right in feeling this convention of notation is backwards or there is some logic to it I am missing that can be explained to me that aren't going to fly over my head.
Thank you
EDIT: A follow up question. One way I thought of to avoid confusion was using the upside down V symbol for cross products. Is that opening up a new worms due to conflict with other uses of that symbol?
2nd Edit: I don't understand any of the other questions on the frontpage - i have come to the right place with people who do extreme mathematics
 A: In mathematics, a symbol means whatever the author says it means. Some notation is standard, but even constructs that do have standard notation can be written in different ways. If $u$ and $v$ are two vectors, then all of the following things could signify their dot product:
$$u\cdot v$$
$$(u,v)$$
$$(u|v)$$
$$\langle u,v\rangle$$
$$\langle u|v\rangle$$
Cross product is usually only written in one way; compared to the dot product, the applications of cross product are rather limited. But I can think of at least one other way to write it:
$$[u,v]$$
(this would be in the case we are considering the cross product as the bracket operation in a Lie algebra).
You are perfectly welcome to use $\times$ for dot product and $\wedge$ for cross product in your own notes or in any exposition you write, but in a classroom it probably won't go over well because you are usually expected to use the notation the instructor taught you to use.
As a side note, $\wedge$ does have a standard meaning: it usually means "and" in formal logic.
A: Perhaps precisely because it is not true that $\times$ is the universal symbol for (scalar) multiplication. I live in Continental Europe and have grown up with the $\cdot$ symbol for multiplication. I have never used $\times$ for scalar mulitplication or seen it outside American textbooks, and in fact my impession is that $\cdot$ is most common in non-English speaking countries. So it seems likely that the notations of the dot and cross products come from Europe.
