Vector Dimensionality I'm sure this is a pretty basic concept, but I'd like to clarify the following.
Assuming $\vec{v} = (v_{1}, \ldots, v_{n})$, for $n > 1$; naturally, $\vec{v}$ has a "presence" (or coordinates) in $n$ dimensions. However, regardless of $n$, as far as I'm aware, $\vec{v}$ has only "measure" in one dimension (that of length) defined by $||\vec{v}||$.
For example, for $\vec{v} = (v_{1}, v_{2}, v_{3})$, we can define the length as $||\vec{v}||$, but, there are no analogous measures for notions of width, height or similar for dimensions two and three. Is this correct?
Given the above, is $\vec{v}$ truly an $n$-dimensional structure or, regardless of $n$, is it always a one-dimensional structure with a mere "presence" in $n$-dimensions?
 A: You can think of a vector $v\in \mathbb R^n$ as prescribing a direction in the space $\mathbb R^n$ together with a magnitude. So, the vector is saying "that way, and at this speed". As such, a vector does not have dimensions. The ambient space $\mathbb R^n$ does. A vector's magnitude, $\|v\|$, can be thought of as the length of the vector. Indeed, vectors in $\mathbb R^n$ can be represented by line segments, and thus are represented by one-dimensional fragments of the ambient space $\mathbb R^n$. 
So, a vector $v\in \mathbb R^n$ is not an $n$-dimensional structure. It resides in a space of dimension $n$, and describes in there a direction + magnitude.
A: Vectors do not have a dimension, only vector spaces do. And we can only discuss length after we have introduced some way to measure it - a norm (or an inner product). And even for $\mathbb{R}^n$ we have a choice of norms to use.
There are many vector spaces for which there is no useful norm.
Basically concepts like length and direction arise in some applications of vector spaces. In other applications, they may be useless.
