# Vector notation - form versus vector?

## Main Question or Discussion Point

Hi there,

During a recent tutorial, I asked my tutor about the notation he uses for vectors - he draws the little half arrow above them and I was curious whether that was significant, as opposed to just underlining vectors. He said it was a mathematical technicality and suggested I look up "forms versus vectors" on Google. However, I haven't been able to find anything conclusive. Could anyone explain the difference between forms and vectors, and the difference in notation?

Thanks!

## Answers and Replies

D H
Staff Emeritus
It's just his convention. Other people have other conventions. There is no standardized way to represent vectors. In print you'll often see vectors as bold upright font, scalars as a plain italic font: ##\mathbf{v}## versus ##v##. Given the shortage of bold upright pieces of chalk (or magic marker), people have to use some other convention when writing on the chalkboard or whiteboard. That's when you'll see stuff like ##\vec v##, or ##\tilde v##, or whatever.

Since there is no standard convention, you'll just have to live with the fact that different people use different conventions.

SteamKing
Staff Emeritus
Homework Helper
Underlining to indicate vectors might get confused with underlining for emphasis. I don't recall seeing this notation used very often (if at all).

D H
Staff Emeritus
Part of what the instructor is getting at is (I think) the difference between a vector and a tensor. In the most abstract, a vector is a member of a vector space. The rules are pretty simple: Vectors can be added associatively and commutatively, there's a zero vector, vectors have additive inverses, and vectors can be multiplied by a scalar. Not one word about how they transform (that concept is key to one-forms, and to tensors in general).

There's not even one word in that abstract concept of vectors and vector spaces about vectors being something with a direction and a magnitude. So even the notation ##\vec v## is a bit misleading for vectors. Some but not all vectors can be characterized as having a direction and a magnitude. That concept pertains to normed vector spaces, but not to vector spaces in general.

pasmith
Homework Helper
I've only ever seen $\mathbf{v}$ or $\vec v$ or $\underline{v}$ (which is only ever used in manuscript, and is an instance of a general convention that what is underlined in manuscript should be in boldface when typeset) used for members of $\mathbb{R}^n$ and on occasion $\mathbb{C}^n$ (or used for functions whose codomains are $\mathbb{R}^n$ or $\mathbb{C}^n$), both of which are equipped with "standard" operations, bases, inner products and norms. For any other vector space - such as the space of functions from an arbitrary non-empty set to $\mathbb{R}$ under pointwise addition and scalar multiplication - no special font or symbol is used.

In the other direction, I've seen plenty of texts which don't use special fonts or symbols for vectors.

Mark44
Mentor
Underlining to indicate vectors might get confused with underlining for emphasis. I don't recall seeing this notation used very often (if at all).
The only place I've seen underlining used for vectors is here. If someone goes to the effort of writing LaTeX code to underline a vector, I wonder why they didn't use the arrow above notation, as in ##\vec{v}##. The underline business for a vector just seems ugly to me.

D H
Staff Emeritus
There are so many ways to represent that something is a vector:
$$\vec{v} \quad \overset{\rightharpoonup}{v} \quad \tilde{v} \quad \bar{v} \quad \underline{v} \quad \mathbf{v} \quad {\frak{v}} \quad \text{and of course} \quad v$$

Mark44
Mentor
There are so many ways to represent that something is a vector:
$$\vec{v} \quad \overset{\rightharpoonup}{v} \quad \tilde{v} \quad \bar{v} \quad \underline{v} \quad \mathbf{v} \quad {\frak{v}} \quad \text{and of course} \quad v$$
You're just showing off :tongue:

jedishrfu
Mentor
You're just showing off :tongue:
You floks forgot one:

The R vector...