6.2 Geometric interpretation of the outer product

The outer product is a powerful tool in geometric algebra, transforming vectors into higher-dimensional objects. It creates bivectors from two vectors, representing oriented plane elements, and trivectors from three vectors, representing oriented volume elements. These new objects encode both magnitude and orientation.

Understanding the outer product is crucial for grasping how geometric algebra represents and manipulates spatial relationships. It's widely used in physics and engineering, from calculating moments of force to representing electromagnetic fields. This geometric interpretation provides a more intuitive way to work with complex spatial concepts.

Geometric Interpretation of the Outer Product

Outer Product of Two Vectors (Bivectors)

• The outer product of two vectors results in a bivector, which represents an oriented plane element in space
• The resulting bivector is perpendicular to both input vectors, forming an oriented plane (e.g., the xy-plane, yz-plane, or xz-plane)
• The magnitude of the resulting bivector is equal to the area of the parallelogram formed by the input vectors
  • Calculated as the product of the magnitudes of the input vectors and the sine of the angle between them
  • For orthogonal vectors, the magnitude is simply the product of the vector magnitudes
• The orientation of the resulting bivector is determined by the order of the input vectors in the outer product, following the right-hand rule (e.g., $\mathbf{e}_1 \wedge \mathbf{e}_2 = \mathbf{e}_1\mathbf{e}_2$)

Outer Product of Three Vectors (Trivectors)

• The outer product of three vectors results in a trivector, which represents an oriented volume element in space
• The resulting trivector is a pseudoscalar, which can be visualized as an oriented volume element (e.g., a cube or parallelepiped)
• The magnitude of the resulting trivector is equal to the volume of the parallelepiped formed by the input vectors
  • Calculated as the scalar triple product of the input vectors
  • For orthogonal vectors, the magnitude is the product of the vector magnitudes
• The orientation of the resulting trivector is determined by the order of the input vectors in the outer product, following the right-hand rule (e.g., $\mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_3$)

Input Vectors vs Bivectors/Trivectors

Relationship between Input Vectors and Resulting Bivectors

• The bivector resulting from the outer product of two vectors is perpendicular to both input vectors, forming an oriented plane
• The geometric relationship between the input vectors and the resulting bivector can be understood using the concept of the parallelogram formed by the input vectors
  • The bivector represents the oriented area of the parallelogram
  • The direction of the bivector is determined by the right-hand rule applied to the order of the input vectors
• Changing the order of the input vectors changes the orientation of the resulting bivector (e.g., $\mathbf{a} \wedge \mathbf{b} = -\mathbf{b} \wedge \mathbf{a}$)

Relationship between Input Vectors and Resulting Trivectors

• The trivector resulting from the outer product of three vectors is a pseudoscalar, which can be visualized as an oriented volume element
• The geometric relationship between the input vectors and the resulting trivector can be understood using the concept of the parallelepiped formed by the input vectors
  • The trivector represents the oriented volume of the parallelepiped
  • The orientation of the trivector is determined by the right-hand rule applied to the order of the input vectors
• Changing the order of the input vectors changes the orientation of the resulting trivector (e.g., $\mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c} = -\mathbf{a} \wedge \mathbf{c} \wedge \mathbf{b}$)

Orientation and Magnitude Encoding

Orientation Encoding in the Outer Product

• The outer product encodes the orientation of the resulting bivector or trivector through the order of the input vectors
• The orientation is determined by the right-hand rule applied to the order of the input vectors
  • For bivectors, the right-hand rule determines the direction of the normal vector to the oriented plane
  • For trivectors, the right-hand rule determines the handedness of the oriented volume element
• Swapping the order of the input vectors changes the orientation of the resulting bivector or trivector (e.g., $\mathbf{a} \wedge \mathbf{b} = -\mathbf{b} \wedge \mathbf{a}$)

Magnitude Encoding in the Outer Product

• The outer product encodes the magnitude of the resulting bivector or trivector through the geometric properties of the input vectors
• For bivectors, the magnitude is equal to the area of the parallelogram formed by the input vectors
  • Calculated as the product of the magnitudes of the input vectors and the sine of the angle between them ($|\mathbf{a} \wedge \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$)
  • For orthogonal vectors, the magnitude is simply the product of the vector magnitudes
• For trivectors, the magnitude is equal to the volume of the parallelepiped formed by the input vectors
  • Calculated as the scalar triple product of the input vectors ($|\mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c}| = |(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}|$)
  • For orthogonal vectors, the magnitude is the product of the vector magnitudes

Applications of the Outer Product in Physics and Engineering

Moment of a Force

• The outer product can be used to calculate the moment of a force about a point
• The bivector represents the oriented area of the parallelogram formed by the force vector and the position vector
  • The magnitude of the bivector is equal to the magnitude of the moment
  • The orientation of the bivector indicates the direction of the moment according to the right-hand rule
• Example: $\mathbf{M} = \mathbf{r} \wedge \mathbf{F}$, where $\mathbf{M}$ is the moment bivector, $\mathbf{r}$ is the position vector, and $\mathbf{F}$ is the force vector

Electromagnetic Field Tensor

• In electromagnetic theory, the outer product of the electric and magnetic field vectors results in a bivector that represents the electromagnetic field tensor
• The electromagnetic field tensor encodes information about the orientation and magnitude of the electromagnetic field
  • The electric field components form the diagonal elements of the tensor
  • The magnetic field components form the off-diagonal elements of the tensor
• The outer product provides a compact way to represent the electromagnetic field tensor and its properties (e.g., $\mathbf{F} = \mathbf{E} \wedge \mathbf{e}_t + \mathbf{B}$, where $\mathbf{F}$ is the electromagnetic field tensor, $\mathbf{E}$ is the electric field vector, $\mathbf{e}_t$ is the time basis vector, and $\mathbf{B}$ is the magnetic field bivector)

Angular Momentum

• The outer product can be used to calculate the angular momentum of a system of particles
• The bivector represents the oriented area swept out by the position and momentum vectors of each particle
  • The magnitude of the bivector is equal to the magnitude of the angular momentum
  • The orientation of the bivector indicates the direction of the angular momentum according to the right-hand rule
• Example: $\mathbf{L} = \mathbf{r} \wedge \mathbf{p}$, where $\mathbf{L}$ is the angular momentum bivector, $\mathbf{r}$ is the position vector, and $\mathbf{p}$ is the momentum vector

Object Orientation in Computer Graphics and Robotics

• The outer product can be used to represent the orientation of objects in space, such as the orientation of a camera or the pose of a robot arm
• Bivectors and trivectors can be used to represent rotations and orientations in 3D space
  • Bivectors represent rotations in a plane (e.g., rotations about the x, y, or z axes)
  • Trivectors represent general rotations in 3D space (e.g., a combination of rotations about multiple axes)
• The outer product provides a compact and intuitive way to represent and manipulate object orientations in computer graphics and robotics applications (e.g., $\mathbf{R} = \exp(\frac{1}{2}\theta\mathbf{n})$, where $\mathbf{R}$ is the rotation operator, $\theta$ is the rotation angle, and $\mathbf{n}$ is the unit bivector representing the rotation plane)