If we interpret a list of two numbers as displacement in a plane, and a list of three numbers as displacement in a space, what of a list of $n$ numbers as displacement in $\R ^n$?

A real vector (or vector, $n$-dimensional vector, $n$-vector) is a length-$n$ list of real numbers.

For $x, y \in \R ^n$, we define the real vector sum (or sum) of $x$ and $y$ as the vector $z \in \R ^n$ where $z_i = x_i + y_i$ for $i = 1, \dots , n$. As usual, we denote the sum by $x+y$, so

\[ x + y = (x_1 +y_1, \dots , x_n + y_n). \]

For $\alpha \in \R $ and $x \in \R ^n$, real scalar-vector product (or scalar product, product) $z \in \R ^{n}$ is defined by $z_i = \alpha x_i$ for $i = 1, \dots , n$. As usual, we denote the product $\alpha x$, and write\[ \alpha x = (\alpha x_1, \dots , \alpha x_n). \]

Our intuition for both of these
*operations* comes from their special cases
in $\R ^2$ and $\R ^3$.
As usual, the real-vector
difference (or
difference) of $x$ and
$y$ is the vector $z \in \R ^{n}$ defined by
$z_i = x_i - y_i$ for $i = 1, \dots , n$.
As usual, we denote it by $x - y$, and note
that $x - y = x + (-y)$.

The algebra given here for vectors is natural
in view of their generalization as
$n$-dimensional *displacements*.
However, we keep in mind that this algebra is
over lists of numbers, and that these sums and
products can be defined on these lists of
numbers regardless of interpretation.