6.1 Definition and properties of the outer product

The outer product is a key operation in geometric algebra, extending the vector cross product to higher dimensions. It takes two vectors and produces a bivector, representing the oriented area spanned by those vectors. This operation is crucial for understanding multidimensional geometry and algebraic structures.

Unlike the inner product, the outer product is anticommutative and grade-increasing. It results in a bivector rather than a scalar, capturing the "independence" of vectors. The outer product's properties, including its null square property and relation to the geometric product, make it essential for advanced geometric calculations.

Outer product definition and notation

Definition and geometric interpretation

• The outer product is a binary operation in geometric algebra that takes two vectors as input and produces a new geometric object called a bivector
• It extends the concept of the vector cross product to higher dimensions, representing the oriented area formed by the parallelogram spanned by the two input vectors
• Given vectors $a$ and $b$, their outer product is written as $a \wedge b$, where the symbol $\wedge$ denotes the outer product operation

Non-commutativity and order dependence

• The outer product is not commutative, meaning that $a \wedge b \neq b \wedge a$
• Swapping the order of the vectors changes the sign of the resulting bivector, as the orientation of the spanned area is reversed
• The order of the vectors in the outer product is crucial for determining the orientation of the resulting bivector (clockwise or counterclockwise)

Properties of the outer product

Anticommutativity and associativity

• The outer product is anticommutative: $a \wedge b = -b \wedge a$, swapping the order of the vectors negates the result
• It is associative: $(a \wedge b) \wedge c = a \wedge (b \wedge c)$, parentheses can be placed around any pair of vectors without changing the result
• The outer product is distributive over addition: $a \wedge (b + c) = a \wedge b + a \wedge c$ and $(a + b) \wedge c = a \wedge c + b \wedge c$

Null square property and parallel vectors

• The outer product of a vector with itself is always zero: $a \wedge a = 0$, known as the null square property
• The outer product of two parallel vectors is zero, as they do not span any area (no parallelogram is formed)
• For non-zero results, the input vectors must be linearly independent (not parallel or antiparallel)

Grade-increasing nature

• The outer product is grade-increasing, meaning that the result has a higher grade than the input vectors
• The outer product of two vectors (grade 1) produces a bivector (grade 2), representing an oriented plane
• In higher dimensions, the outer product of k vectors produces a k-vector (grade k), representing an oriented k-dimensional subspace

Computing the outer product

General procedure and basis vector rules

• To compute the outer product of vectors $a = a_1e_1 + a_2e_2 + ... + a_ne_n$ and $b = b_1e_1 + b_2e_2 + ... + b_ne_n$:
  1. Multiply each component of $a$ with each component of $b$
  2. Combine the results using the outer product of the basis vectors
• The outer product of basis vectors follows specific rules: $e_i \wedge e_j = -e_j \wedge e_i$ for $i \neq j$, and $e_i \wedge e_i = 0$

Examples in 2D and 3D geometric algebra

• In 2D geometric algebra, for $a = a_1e_1 + a_2e_2$ and $b = b_1e_1 + b_2e_2$:
  • $a \wedge b = (a_1b_2 - a_2b_1) e_1 \wedge e_2$
  • The result is a scalar multiple of the bivector $e_1 \wedge e_2$, representing the oriented area in the $e_1e_2$ plane
• In 3D geometric algebra, for $a = a_1e_1 + a_2e_2 + a_3e_3$ and $b = b_1e_1 + b_2e_2 + b_3e_3$:
  • $a \wedge b = (a_1b_2 - a_2b_1) e_1 \wedge e_2 + (a_1b_3 - a_3b_1) e_1 \wedge e_3 + (a_2b_3 - a_3b_2) e_2 \wedge e_3$
  • The result is a linear combination of the bivectors $e_1 \wedge e_2$, $e_1 \wedge e_3$, and $e_2 \wedge e_3$, representing the oriented areas in the respective planes

Geometric interpretation of the result

• The resulting bivector from the outer product represents the oriented area spanned by the two input vectors
• The orientation (clockwise or counterclockwise) is determined by the order of the vectors in the outer product
• The magnitude of the bivector corresponds to the area of the parallelogram formed by the input vectors

Outer product vs inner product

Commutativity and anticommutativity

• The inner product is commutative: $a \cdot b = b \cdot a$, the order of the vectors does not matter
• The outer product is anticommutative: $a \wedge b = -b \wedge a$, swapping the order of the vectors negates the result

Scalar vs bivector results

• The inner product results in a scalar value, representing the projection of one vector onto another
• The outer product results in a bivector, representing the oriented area spanned by the two vectors
• The inner product measures "similarity" between vectors, while the outer product captures their "difference" or "independence"

Grade-preserving vs grade-increasing

• The inner product is grade-preserving, the result has the same grade as the input vectors (grade 0 for vectors)
• The outer product is grade-increasing, producing a result with a higher grade than the input vectors (bivector for two vectors)

Self-inner and self-outer products

• The inner product of a vector with itself is always non-negative: $a \cdot a \geq 0$, equal to the squared magnitude of the vector
• The outer product of a vector with itself is always zero: $a \wedge a = 0$, due to the null square property

Relation to the geometric product

• The geometric product, another fundamental operation in geometric algebra, can be expressed as the sum of the outer product and the inner product: $ab = a \cdot b + a \wedge b$
• The inner product (scalar part) and outer product (bivector part) can be extracted from the geometric product using grade projection operators