Geometric Product
- defines the multiplication of vectors that results in higher-dimensional objects called multivectors
- geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra
- can be thought of as the “reduced form” of the outer product
Geometric Product - Definition/Assumptions
For vectors 𝑣,𝑢,𝑤, the geometric product on vectors is defined as follows:
- Associativity:
- (𝑢𝑣)𝑤 = 𝑢(𝑣𝑤)
- Left and right distributivity over addition:
- 𝑣(𝑢 + 𝑤) = 𝑣𝑢 + 𝑣𝑤
- (𝑢 + 𝑤)𝑣 = 𝑢𝑣 + 𝑤𝑣
- Contraction:
- 𝑣2= 𝒬(𝑣) = 𝜀𝑣|𝑣|2
- where:
- 𝒬 is the quadratic form
- |𝑣| is the magnitude of 𝑣
- 𝜀𝑣 is the metric signature. For a space with Euclidean metric 𝜀𝑣 is 1 so can be omitted, and the contraction condition becomes: 𝑣2= ||𝑣||2
The contraction assumption also yields the following identities:
-
the product of a basis vector with itself is 1: 𝑒𝑖𝑒𝑖 = 1
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2= ||𝑣||2
𝑣
- 𝑒𝑖2 = ||𝑒𝑖||2
- 𝑒𝑖2 = 12
- 𝑒𝑖2 = 1
- 𝑒𝑖𝑒𝑖 = 1
-
basis vectors anti-commute with other basis vectors: 𝑒𝑖𝑒𝑗 = −𝑒𝑗𝑒𝑖
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-
||𝑒𝑖 + 𝑒𝑗|| = sqrt(12 + 12) = sqrt(2)
2= ||𝑣||2
𝑣
- (𝑒𝑖 + 𝑒𝑗)2 = sqrt(2)2
- 𝑒𝑖𝑒𝑖 + 𝑒𝑖𝑒𝑗+ 𝑒𝑗𝑒𝑖 + 𝑒𝑗𝑒𝑗 = 2
- 1 + 𝑒𝑖𝑒𝑗+ 𝑒𝑗𝑒𝑖 + 1 = 2
- 𝑒𝑖𝑒𝑗+ 𝑒𝑗𝑒𝑖 = 0
- 𝑒𝑖𝑒𝑗 = -𝑒𝑗𝑒𝑖
-
Geometric Product of 1-Vectors
The geometric product between two 1-vectors 𝑣,𝑢 equals the sum of the dot product(⋅) and wedge product (∧)
- 𝑢𝑣 = 𝑢⋅𝑣 + 𝑢∧𝑣
Derivation
derivation
Given 2 vectors:
- 𝑢 = 𝑢1𝑒1 + 𝑢2𝑒2+ …
- 𝑣 = 𝑣1𝑒1 + 𝑣2𝑒2+ …
The geometric product is distributive over addition:
- 𝑢𝑣 = 𝑢(𝑣1𝑒1) + 𝑢(𝑣2𝑒2) + …
- 𝑢𝑣 = (𝑢1𝑒1 + 𝑢2𝑒2+ …)(𝑣1𝑒1) + (𝑢1𝑒1 + 𝑢2𝑒2+ …)(𝑣2𝑒2) + …
- 𝑢𝑣 = (𝑢1𝑒1)(𝑣1𝑒1) + (𝑢2𝑒2)(𝑣1𝑒1) + … + (𝑢1𝑒1)(𝑣2𝑒2) + (𝑢2𝑒2)(𝑣2𝑒2) + …
The geometric product is associative:
- 𝑢𝑣 = 𝑢1𝑒1𝑣1𝑒1 + 𝑢2𝑒2𝑣1𝑒1 + … + 𝑢1𝑒1𝑣2𝑒2 + 𝑢2𝑒2𝑣2𝑒2 + …
The geometric product has the following identities:
- scalars commute with basis vectors: 𝛼𝑒𝑖 = 𝑒𝑖𝛼
- the product of a basis vector with itself is 1: 𝑒𝑖𝑒𝑖 = 1
- basis vectors anti-commute with other basis vectors: 𝑒𝑖𝑒𝑗 = −𝑒𝑗𝑒𝑖
Continuing our computation:
- 𝑢𝑣 = 𝑢1𝑣1𝑒1𝑒1 + 𝑢2𝑣1𝑒2𝑒1 + … + 𝑢1𝑣2𝑒1𝑒2+ 𝑢2𝑣2𝑒2𝑒2 + …
- 𝑢𝑣 = 𝑢1𝑣1 - 𝑢2𝑣1𝑒1𝑒2 + … + 𝑢1𝑣2𝑒1𝑒2+ 𝑢2𝑣2 + …
- 𝑢𝑣 = 𝑢1𝑣1 + 𝑢2𝑣2 + … + (𝑢1𝑣2 - 𝑢2𝑣1)𝑒1𝑒2 + …
- 𝑢𝑣 = 𝑢⋅𝑣 + (𝑢1𝑣2 - 𝑢2𝑣1)𝑒1𝑒2 + … # by definition of dot product
- 𝑢𝑣 = 𝑢⋅𝑣 + 𝑢∧𝑣 # by definition of wedge product
The geometric product between two 1-vectors 𝑣,𝑢 equals the sum of the symmetric product and an antisymmetric product
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