bivector

A bivector, or an oriented area , is an oriented segment of a plane.

The area of a bivector is called its norm, and is denoted by . ¹

A bivector can be created from the wedge product . Geometrically, such a bivector represents the parallelogram formed by the vectors    and  , in the plane they span together. ²

Just like vectors that shares the same line, coplanar bivectors with the same norm are equal. In other word, a bivector has no information of its location. ¹

Bivector Vector Space

Just like the Euclidean vectors , Bivectors forms a vector space, with the scalar multiplication multiplies their area.

For parallel bivectors, vector addition just sum their areas. For non-parallel bivectors, vector addition gives back the result of something like following

In 2D

In 2D, there is only one plane, the plane. So a 2D bivector only has one component . The norm is the the signed area of the parallelogram the two vectors form together.

In 3D

In 3D, a bivector contains 3 orthogonal basis planes: , , .

Just like the coordinates of a vector can be view as the projections of the vector onto the three orthogonal basis axes, the coordinates of a bivector can be thought of as projections of the bivector plane onto the three orthogonal basis planes:

Related

• Euclidean vectors - Represents oriented lengths
• trivector - Represents oriented volumes

Footnotes

1. Linear and Geometric Algebra 5.1 ↩ ↩²

2. Let’s remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra) - Marc ten Bosch ↩