Definition:Vector/Real Euclidean Space

A vector is defined as element of a vector space.

We have that $\R^n$ is a vector space.

Hence a vector in $\R^n$ is defined as any element of $\R^n$.

Note that such a vector is so commonly used that sometimes the term vector is implied to mean "a vector in $\R^n$".

R1
As the Reals form a vector space, every real number is a vector.

Note that every real number is also a scalar.

Hence a vector in $\R^n$ is sometimes imprecisely used to mean "a vector in $\R^n$, $n > 1"$.

Geometric Interpretation
From Real Number Line, the real number space is often represented by an infinite straight line.

By the same token, a vector in $\R$ can be represented by a directed line segment.

Formally, a vector $\langle{x_1}\rangle, \ x_1 \in \R$ is accurately represented by the set of all directed line segments having:


 * Magnitude $\left \vert{x_1}\right \vert$


 * Direction dependent on whether $x_1 < 0$ or $x_1 > 0$

By convention, if only one axis is under consideration, the line is placed horizontally, such that segmented oriented towards the right are positive, to the left negative.

Note that in such a context the zero vector can be interpreted as a directed line segment beginning and terminating at the same point.

Comment
The reader should be aware that a vector in $\R^n$ is and only is an ordered $n$-tuple of $n$ real numbers. The geometric interpretations given above are only representations of vectors.

Further, the geometric interpretation of a vector is accurately described as the set of all  line segments equivalent to a given directed line segment, rather than any particular line segment.