k-blades vs k-vectors vs multivectors

• blades are the combined geometric products between scalars and vectors of an orthonormal basis in the associated vector space
• a blade’s grade is the number of basis vectors that are multiplied together (after it is reduced)
  • the geometric product of >𝑛 vectors can always be reduced to a geometric product with ≤𝑛 vectors, where 𝑛 is the dimension of the associated vector space
• a blade of grade k is called a k-blade
• permutations of the same product end up being scaled versions of each other, so it is possible to define a set of blades as a basis for the vector space of multivectors by choosing one of each permutation set
  • the standard basis of ℝ³ {𝑒₁, 𝑒₂, 𝑒₃} leads to this standard basis of blades for 𝔾³ {1, 𝑒₁, 𝑒₂, 𝑒₃, 𝑒₁𝑒₂, 𝑒₁𝑒₃, 𝑒₂𝑒₃, 𝑒₁𝑒₂𝑒₃}
• a multivector is defined as a linear combination of possibly-different-grade k-blades (e.g. such as the summation of a scalar, a vector, and a 2-vector)
• a k-vector is defined as a linear combination of same-grade k-blades (thus; a homogeneous multivector)

If a given element is homogeneous of a grade k, then it is a k-vector, but not necessarily a k-blade. Such an element is a k-blade when it can be expressed as the exterior product of k vectors. A geometric algebra generated by a 4-dimensional vector space illustrates the point with an example: The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade. In a geometric algebra generated by a vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.

Examples

Here are some examples of each term, using the standard basis of ℝ³ as the defining vectors. The second examples are not in reduced form. To reduce them, we apply the following identities that define the geometric product:

• scalars commute with basis vectors: 𝛼𝑒_𝑖= 𝑒_𝑖𝛼
• the product of a basis vector with itself is 1: 𝑒_𝑖𝑒_𝑖= 1
• basis vectors anti-commute with other basis vectors: 𝑒_𝑖𝑒_𝑗= −𝑒_𝑗𝑒_𝑖