Vectors

The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive real number a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction, instead.

The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (right image below). Equivalently 2w is the sum w + w. Moreover, (?1)v = ?v has the opposite direction and the same length as v (blue vector pointing down in the right image).
Vector addition: the sum v + w (black) of the vectors v (blue) and w (red) is shown. Scalar multiplication: the multiples ?v and 2w are shown.
Second example: ordered pairs of numbers

A second key example of a vector space is provided by pairs of real numbers x and y. (The order of the components x and y is significant, so such a pair is also called an ordered pair.) Such a pair is written as (x, y). The sum of two such pairs and multiplication of a pair with a number is defined as follows:

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)

and

a?(x, y) = (ax, ay).

The first example above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points.
Definition