Geometric algebra foundations

Grades and wedge

Before a PGA point can be represented as a bivector, grades, blades, multivectors, and wedge products need concrete definitions. Start with ordinary vectors and the directed areas they span.

Exterior algebra is the part of geometric algebra that talks about oriented spans before distance, angle, reflection, or rotation enter the calculation. It answers questions like: what is the oriented line segment described by one vector, what is the oriented area described by two independent vectors, and what happens when the vectors fail to be independent?

The wedge product, written ^, means "span these together." If two vectors point in different directions, they span a directed area. If they are parallel, the span collapses.

The later PGA pages use the same grade vocabulary, but the geometric roles change. In 2D PGA, points are grade-2 bivectors, so this foundation page first builds the grade-2 idea in the simpler exterior-algebra setting.

Grades Are Oriented Measure

A grade says what kind of oriented measure a piece carries. It records the algebraic dimension of the span:

grade 0: scalar      // magnitude with no direction
grade 1: vector      // directed length
grade 2: bivector    // directed area
grade 3: trivector   // directed volume

In ordinary 3D space, the basis vectors are e1, e2, and e3. Wedge two different basis vectors together and you get a basis bivector:

e1 ^ e2 = e12
e1 ^ e3 = e13
e2 ^ e3 = e23

Read these as coordinate-plane area slots. e12 is area in the e1/e2 plane, e13 is area in the e1/e3 plane, and e23 is area in the e2/e3 plane.

Blades And Multivectors

A blade is one simple oriented span. A vector is a blade. The wedge of two independent vectors,a ^ b, is also a blade: one oriented parallelogram. The wedge of three independent vectors, a ^ b ^ c, is one oriented volume element.

A multivector is a sum of grade pieces. It can carry a scalar, vector, bivector, and trivector part side by side:

M = 2 + 3e1 - e2 + 5e12 + e123

Each piece stays in its grade slot. A scalar coefficient and an e12 coefficient are different kinds of information, even though both are numbers.

A sum of same-grade pieces can be more general than a single simple blade. In 3D, every bivector happens to correspond to one plane of area, while in higher dimensions a sum of bivectors can describe a composite area. For the 2D and 3D intuition on this page, keep track of the grade and the basis blade.

Wedge Builds Oriented Spans

If a and b are vectors, thena ^ b is the oriented parallelogram they span. It stores three things at once:

a ^ b
  -> the plane of the span
  -> the amount of oriented area
  -> the orientation from a toward b

In 2D, there is only one coordinate area slot, e12. In 3D, the same bivector can have components in the three coordinate planes:

a ^ b = B12 e12 + B13 e13 + B23 e23

Those components are coordinates of one oriented area element, the same way a vector's e1, e2, and e3 components are coordinates of one directed length.

Orientation Is Part Of The Object

Order matters. Swapping the vectors flips the orientation of the span:

b ^ a = -(a ^ b)

The sign records traversal direction. The bivector distinguishes clockwise from counterclockwise traversal of the same parallelogram. The spanned plane can stay fixed while the orientation sign changes.

A vector wedged with itself has no area because it repeats the same direction:

a ^ a = 0

When the vectors become parallel, the parallelogram collapses and the bivector components go to zero. Parallel inputs add no independent direction, so the algebra returns the zero bivector.

Compute It Like Algebra

The wedge has a geometric meaning and a component calculation. There is no metric step here. The wedge rules are bilinear distribution, repeated directions going to zero, and sign changes when basis vectors swap order.

(2e1 + e2) ^ (3e2)
= 2e1 ^ 3e2 + e2 ^ 3e2
= 6e12 + 0
= 6e12

The calculation follows the same expand-and-collect structure used by the geometric product, with fewer rules: distribute terms, simplify basis wedges, then collect the remaining bivector components.

Bilinearity Gives Component Formulas

Wedge is bilinear, which means it distributes over vector addition and scalar multiplication in each input:

(a1 + a2) ^ b = a1 ^ b + a2 ^ b
a ^ (b1 + b2) = a ^ b1 + a ^ b2
(ca) ^ b = c(a ^ b)

Bilinearity is why a and b can be expanded into basis-vector terms, every pair can be wedged, and the result can be collected. The component formula is the distributed wedge written compactly:

a = a1 e1 + a2 e2 + a3 e3
b = b1 e1 + b2 e2 + b3 e3

a ^ b =
  (a1 b2 - a2 b1) e12
+ (a1 b3 - a3 b1) e13
+ (a2 b3 - a3 b2) e23

Each coefficient is a signed area projection into a coordinate plane. The minus terms are the orientation flips: e2 ^ e1 = -e12, e3 ^ e1 = -e13, and e3 ^ e2 = -e23.

Adding Bivectors Adds Area Components

Adding two bivectors means adding like basis-plane components. If one wedge contributes2e12 + e13 and another contributes -5e12 + 3e23, their sum is:

(2e12 + e13) + (-5e12 + 3e23) = -3e12 + e13 + 3e23

The result is still grade 2 because every surviving term is a bivector term. The coefficients changed while the grade stayed fixed.

This is one reason multivectors are useful for graphics. A scalar measurement, a vector direction, and a bivector span can live in one algebraic object without pretending they are the same kind of coordinate.

How This Carries Into PGA

In ordinary Euclidean exterior algebra, a bivector is easiest to picture as directed area. In the 2D PGA pages, a finite point is also represented by a grade-2 bivector:

P = e12 - x e02 + y e01

The projective model uses a grade-2 slot for points. The grade vocabulary survives, while the geometric interpretation changes because the basis and metric have changed.

The next foundation page adds the geometric product. That product keeps the exterior-algebra practice of oriented basis blades, then adds metric rules so repeated basis vectors can simplify to scalar values. With those rules, multivectors can express reflection, rotation, and PGA motors.