What is Geometric Algebra?

Geometric Algebra (GA) is a powerful mathematical framework that unifies linear algebra, vector calculus, and complex numbers. It simplifies geometric transformations, rotations, and higher-dimensional operations, making it highly applicable in fields like physics, computer graphics, and robotics.

1. Rotation in 3D Using Rotors

Rotors provide an efficient way to handle 3D rotations. Here’s an example:

Problem:

Rotate the vector v = e₁ + 2e₂ by 90° around the e₃-axis.

Solution:

The rotor is computed as:

R = cos(π/4) + (e₁ ∧ e₂)sin(π/4)

The rotated vector is:

v' = -2e₁ + e₂

2. Computing the Area of a Triangle

Problem:

Find the area of a triangle with vertices A(0,0), B(2,0), and C(0,3).

Solution:

The wedge product of two edge vectors gives the area of the parallelogram. For the triangle, divide by 2:

Area = ½ × |u ∧ v| = 3

3. Maxwell’s Equations in Physics

In GA, Maxwell’s equations for electromagnetism are compactly represented as:

∂F = J

Here, F is the electromagnetic field bivector, and J is the current density. This replaces the traditional four equations, streamlining physics computations.

4. Intersection of Two Planes

Problem:

Find the line of intersection between the planes P₁: x + y + z = 0 and P₂: x – y + z = 1.

Solution:

The outer product of the plane bivectors gives the intersection line:

L = P₁ ∧ P₂

5. Reflection of a Vector

Problem:

Reflect the vector v = e₁ + 2e₂ across a = e₁.

Solution:

The reflection formula gives:

v' = e₁ - 2e₂