Primer on Geometric Algebra

Previous section: Rules for Vector Addition.

Next section: Parametric Equations.

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

• Additive and multiplicative identities

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

• Distributive

          α(a + b) = αa + αb

          (α + β)a = αa + βa 

Example: Repeated vector addition as scalar multiplication

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

• Associative

          α(βa) = (αβ)a 

• Commutative

          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

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

          b = βa 

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

          x(α₁,α₂,...,α_n) = α₁a₁ + α₂a₂ + ... + α_na_n

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

Previous section: Rules for Vector Addition.

Next section: Parametric Equations.