Editing Euclidean vector (section)

===Addition and subtraction===
{{Further|Vector space}}
The sum of '''a''' and '''b''' of two vectors may be defined as
<math display=block>\mathbf{a}+\mathbf{b}
=(a_1+b_1)\mathbf{e}_1
+(a_2+b_2)\mathbf{e}_2
+(a_3+b_3)\mathbf{e}_3.</math>
The resulting vector is sometimes called the '''resultant vector''' of '''a''' and '''b'''.

The addition may be represented graphically by placing the tail of the arrow '''b''' at the head of the arrow '''a''', and then drawing an arrow from the tail of '''a''' to the head of '''b'''. The new arrow drawn represents the vector '''a''' + '''b''', as illustrated below:<ref name=":2" />

[[Image:Vector addition.svg|class=skin-invert-image|250px|center|The addition of two vectors '''a''' and '''b''']]

This addition method is sometimes called the ''parallelogram rule'' because '''a''' and '''b''' form the sides of a [[parallelogram]] and '''a''' + '''b''' is one of the diagonals. If '''a''' and '''b''' are bound vectors that have the same base point, this point will also be the base point of '''a''' + '''b'''. One can check geometrically that '''a''' + '''b''' = '''b''' + '''a''' and ('''a''' + '''b''') + '''c''' = '''a''' + ('''b''' + '''c''').

The difference of '''a''' and '''b''' is

<math display=block>\mathbf{a}-\mathbf{b}
=(a_1-b_1)\mathbf{e}_1
+(a_2-b_2)\mathbf{e}_2
+(a_3-b_3)\mathbf{e}_3.</math>

Subtraction of two vectors can be geometrically illustrated as follows: to subtract '''b''' from '''a''', place the tails of '''a''' and '''b''' at the same point, and then draw an arrow from the head of '''b''' to the head of '''a'''. This new arrow represents the vector '''(-b)''' + '''a''', with '''(-b)''' being the opposite of  '''b''',  see drawing. And '''(-b)''' + '''a''' = '''a''' − '''b'''.

[[Image:Vector subtraction.svg|class=skin-invert-image|125px|center|The subtraction of two vectors '''a''' and '''b''']]