Editing Euclidean vector (section)

===Scalar multiplication===
{{main|Scalar multiplication}}

[[Image:Scalar multiplication by r=3.svg|class=skin-invert-image|250px|thumb|right|Scalar multiplication of a vector by a factor of 3 stretches the vector out.]]
A vector may also be multiplied, or re-''scaled'', by any [[real number]] ''r''. In the context of [[vector analysis|conventional vector algebra]], these real numbers are often called '''scalars''' (from ''scale'') to distinguish them from vectors. The operation of multiplying a vector by a scalar is called ''scalar multiplication''. The resulting vector is

<math display=block>r\mathbf{a}=(ra_1)\mathbf{e}_1
+(ra_2)\mathbf{e}_2
+(ra_3)\mathbf{e}_3.</math>

Intuitively, multiplying by a scalar ''r'' stretches a vector out by a factor of ''r''. Geometrically, this can be visualized (at least in the case when ''r'' is an integer) as placing ''r'' copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.

If ''r'' is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (''r'' = −1 and ''r'' = 2) are given below:

[[Image:Scalar multiplication of vectors2.svg|class=skin-invert-image|250px|thumb|left|The scalar multiplications −'''a''' and 2'''a''' of a vector '''a''']]

Scalar multiplication is [[Distributivity|distributive]] over vector addition in the following sense: ''r''('''a''' + '''b''') = ''r'''''a''' + ''r'''''b''' for all vectors '''a''' and '''b''' and all scalars ''r''. One can also show that '''a''' − '''b''' = '''a''' + (−1)'''b'''.
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The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a [[vector space]]. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated.
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