Definition:Vector/Real Euclidean Space

Definition
A vector is defined as an element of a vector space.

We have that $\R^n$, with the operations of vector addition and scalar multiplication, form a real vector space.

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

$\R^1$
As $\R$ forms a vector space, every real number is a vector.

Note that as $\R$ is a vector space over itself, 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 the Cantor-Dedekind Hypothesis, the real number line $\R$ can be 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 $\sequence {x_1}, x_1 \in \R$ is accurately represented by the set of all directed line segments having:


 * Magnitude $\size {x_1}$


 * 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 segments 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.

$\R^2$
We have that $\R^2$ is a vector space.

Hence any ordered $2$-tuple of $2$ real numbers is a vector.

Geometric Interpretation
From the definition of the real number plane, we can represent the vector space $\R^2$ by points on the plane.

That is, every pair of coordinates $\tuple {x_1, x_2}$ can be uniquely defined by a point in the plane.

An arrow with base at the origin and terminal point $\tuple {x_1, x_2}$ is defined to have the length equal to the magnitude of the vector, and direction defined by the relative location of $\tuple {x_1, x_2}$ with the origin as the point of reference.

Each vector is then represented by the set of all directed line segments with:


 * Magnitude $\sqrt {x_1^2 + x_2^2}$


 * Direction equal to the direction of $\overrightarrow {\tuple {0, 0} \tuple {x_1, x_2} }$

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.