Rules for Multiplication by Scalars

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(Scalar variables are typically denoted by Greek letters and/or italics) 

Additive and multiplicative identities

          1a = a,          (1)a = –a,          0a =


          α(a + b) = αa + αb

          (α + β)a = αa + β

Example: Repeated vector addition as scalar multiplication

          a + a + a = (1 + 1 + 1)a = 3


          α(βa) = (αβ)


          αa = aα 

Magnitude and direction: Every vector a has a unique scalar magnitude  a = |a| 

          and (if a 0) a direction (usually represented by a caret, or hat, over the vector) so that

\[\Large{{\bf{a}} = a{\bf{\hat a}}}\]

Collinearity. Nonzero vectors a and b are said to be collinear or linearly dependent if there is a scalar β such that

          b = β

Linear independence. Nonzero vectors a1,a2,...,an are said to be linearly independent if

          x(α1,α2,...,αn) = α1a1α2a2 + ... + αnan

is not zero for any combination of scalars α1,α2,...,αn (not all zero). The scalars {α1,α2,...,αn} are said to be coordinates for the vector x(α1,α2,...,αn) with respect to the basis {a1,a2,...,an}. The set {x(α1,α2,...,αn)} for all values of the coordinates is an n-dimensional vector space.

Previous sectionRules for Vector Addition.

Next section: Parametric Equations.

© David Hestenes 2005, 2014