Why $\vec{r}$ is commonly use for vector equation? I'm wondering why $\vec{r}$ is commonly use in mathematics (vector calculus, line integrals) and physics for denote the vector equation.
Edit/Added clarification: I'm wondering why the letter $r$ is commonly use in mathematics (vector calculus, line integrals) and physics for denote the vector equation.
 A: I have seen the letter $r$ used in physics (usually denoted as $\mathbf{r}$ or less commonly $\vec{r}$) to denote the position vector of a particle. It is also called the radius vector and may be that's why $r$ is more commonly used. In math, when the variable $r$ is used, it's not really for any particular reason. Also I haven't seen $\mathbf{r}$ or $\vec{r}$ used in pure math (probably used more in some applied math fields which are related to physics).
A: When we write $\vec r$, the "arrow" above the variable $r$ helps distinguish the vector $r = \langle x_1, x_2, \cdots, x_n\rangle$ from the $n-$ tuple $r = (x_1, x_2, \cdots, x_n)$, and helps as well to distinguish a variable representing a vector from a variable representing a scalar value.
The use of the variable $r$ to denote (name) a vector is arbitrary.
A: Vectors can be regarded as "arrows" in 3D space. So, we put little arrows over the symbols, to help distinguish vectors from plain numbers. There are several other common notations. Some people like to underline the the symbol, or put a little squiggle under it, and others (like me) just use bold letters.
If you're asking why people use the letter "r" (as opposed to some other letter), I don't know why. Personally, I use "$\mathbf x$" more than "$\mathbf r$".
Maybe people use "$\mathbf r$" because $r$ is a natural symbol for the "radial distance" from the origin to a point. So, if we denote  the corresponding position vector by $\mathbf r$, then we have the nice relationship $r = \Vert \mathbf r \Vert$.
A: Well, it is actually not commonly used in all areas of mathematics. It is used in physics and occasionally in mathematical areas which deal with problems motivated by physics (e.g. differential equations). 
There vectors are most commonly just column vectors or 'arrows' in an n-dimensional space. This is indicated by the arrow on the top. The reason to use this, is to make it easier to distinguish between vectors and scalars.
A: In general, the use of $\vec{r}$  specifically denotes a vector that points from the origin to a position in real space. As opposed to $\vec{v}$, which usually denotes an arbitrary vector that could represent any vector quantity (or in the case of physics, velocity), $\vec{r}$ carries the implication that the vector represents a position in real space.
I believe this comes from the fact that it is a position in real space.
$$\vec{r}\in\mathbb{R}^d$$
It is also a possibiity that it stands for radial vector, because it is a vector that is radial to the origin.
A: It could be because $\vec r \cdot\vec r=|r|^2$ so when one is using polar forms of co-ordinates (natural for forces etc originating at a point source) the designation of the vector as $\vec r$ takes its cue from the magnitude $|r|$.
But there are many situations where other notations are used.
