Geometric Product Visualizer

Explore AB = A·B + A∧B in 2D space

Controls

Vector A

Vector
-5 1.0 5
-5 1.0 5

Vector B

Vector
-5 1.0 5
-5 0.0 5

Geometric Product AB

Dot Product (A·B): 1.0
Wedge Product (A∧B): 1.0
Total (AB): 2.0

About the Geometric Product

The geometric product is a fundamental operation in geometric algebra that combines the dot product (A·B) and wedge product (A∧B):

AB = A·B + A∧B

  • Dot Product (A·B): Represents the symmetric part of the product, giving a scalar value that measures how much the vectors align.
  • Wedge Product (A∧B): Represents the antisymmetric part, giving a bivector that represents the oriented area spanned by the vectors.
  • When A and B are vectors, A·B is a scalar and A∧B is a bivector.
  • When one operand is a bivector, the geometric product includes additional terms that rotate and scale the other operand.

Use the controls to explore how different combinations of vectors and bivectors affect the geometric product.