Editing Vector space (section)

===Arrows in the plane===
<div class=skin-invert-image>{{multiple image
 | total_width = 200
 | direction = vertical
 | image1 = Vector addition3.svg
 | caption1 = Vector addition: the sum {{math|'''v''' + '''w'''}} (black) of the vectors {{math|'''v'''}} (blue) and {{math|'''w'''}} (red) is shown.
 | image2 = Scalar multiplication.svg
 | caption2 = Scalar multiplication: the multiples {{math|−'''v'''}} and {{math|2'''w'''}} are shown.
}}</div>
The first example of a vector space consists of [[arrow (symbol)|arrow]]s in a fixed [[plane (geometry)|plane]], starting at one fixed point. This is used in physics to describe [[force]]s or [[velocity|velocities]].{{sfn|Kreyszig|2020|p=[https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA355 355]}} Given any two such arrows, {{math|'''v'''}} and {{math|'''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 {{math|'''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]] {{math|''a''}}, the arrow that has the same direction as {{math|'''v'''}}, but is dilated or shrunk by multiplying its length by {{math|''a''}}, is called ''multiplication'' of {{math|'''v'''}} by {{math|''a''}}. It is denoted {{math|''a'''''v'''}}. When {{math|''a''}} is negative, {{math|''a'''''v'''}} is defined as the arrow pointing in the opposite direction instead.{{sfn|Kreyszig|2020|p=[https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA358 358&ndash;359]}}

The following shows a few examples: if {{math|1=''a'' = 2}}, the resulting vector {{math|''a'''''w'''}} has the same direction as {{math|'''w'''}}, but is stretched to the double length of {{math|'''w'''}} (the second image). Equivalently, {{math|2'''w'''}} is the sum {{math|'''w''' + '''w'''}}. Moreover, {{math|1=(−1)'''v''' = −'''v'''}} has the opposite direction and the same length as {{math|'''v'''}} (blue vector pointing down in the second image).