Which is more correct, $\| \vec{a} \|$ or $|\vec{a}|$? The question is pretty self explanatory, but I’ve encountered situations where, for the length of some vector $\vec{a}$, to denote the length (or magnitude, which ever you prefer) as either $\| \vec{a}\|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ or $|\vec{a}|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ and I was wondering which notation is more widely accepted, per say? I’ve tried researching this and different websites actually use different notation. 
Any help is appreciated, thank you.
 A: Definitions are not judged "correct" or "incorrect".  Both of your notations are used in various contexts in mathematics.  There is a problem only if the writer and the reader do not understand each other.
A: You see both sets of notation. For vectors you see ${\bf v}$, $\vec{v}$ and $\underline{v}$, perhaps others. For the norm you see $|\cdot|$ and $\| \cdot \|$. It depends if you're in high-school or university, do physics or pure maths. 
As a mathematician, I prefer $\|{\bf v}\|$ for the norm of a vector. By hand, that would be written as $\| \underline{v}\|$, but that's just because it's hard to write bold font by hand. 
(Same reason we use $\mathbb R$ for what was traditionally ${\bf R}$)
I like to use $|\cdot|$ for the modulus function, a.k.a. absolute value, which applies to scalars. It might seem silly when both $|\cdot|$ and $\| \cdot \|$ measure "size" in some way, but it makes it easier for the reader to see what is a vector and what is a scalar. 
(To be technical: A vector space has a set of vectors, and an accompanying scalar field. There is often an idea of "size" in the vector space, and an idea of "size" in the scalar field. I like to use $\| \cdot \|$ for the norm in the vector space and $| \cdot |$ for the norm in the scalar field.)
For example, given two intersecting lines with respective direction vectors ${\bf u}$ and ${\bf v}$, the acute angle of intersection $\theta$ satisfies $|{\bf u} \cdot {\bf v}| = \|{\bf u}\| \|{\bf v}\| \cos\theta$. The scalar/inner/dot product ${\bf u} \cdot {\bf v}$ is a scalar, and so $|{\bf u} \cdot {\bf v}|$ is the absolute value/modulus of that scalar. On the other hand ${\bf u}$ and ${\bf v}$ are vectors and $\|{\bf u}\|$ and $\|{\bf v}\|$ are the norms of those vectors.
I hope this makes sense.
A: I would say, it really depends on the context. Both are widely accepted and understood. However, the nice thing is, that the two notations allow you to distinguish between two notions of "length". So in one extreme, in an introductory course on vectors I would write
$$\|a\| = \sqrt{|a_1|^2+ |a_2|^2+...+|a_n|^2}$$
to explicitely distinguish the notion of absolute value in the real numbers and length of a vector.
In the other extreme, if I am later interested in some basic functional analysis, I would instead use notation
$$\|f\| = \sqrt{\int_{\mathbb{R}^n} |f(x)|^2 dx}$$
for some function $f:\mathbb{R}^n \to \mathbb{R}^n$, where then
$$|f(x)| = \sqrt{f_1(x)^2+...+f_n(x)^2},$$
this time to explicitely distinguish between the $L^2$ norm of a function and the length of its value at $x$.
edit: I did not put vector arrows here, because I personally do not use them, however this answer of course also works with $\vec{a}$ and $\vec{f}$.
A: The first one is the more correct, in the sense that is preferable when you are dealing both with vectors and numbers and it is necessary to avoid misunderstanding.
The notation $|\cdot|$ indicates the absolute value for numbers, but it is also frequently and widely used for vectors when it is clear from the context.
