4.2 Geometric product of vectors and its interpretation

The geometric product of vectors combines the dot and wedge products, offering a powerful tool for manipulating and interpreting vector relationships. It encodes both the magnitude and direction of vector interactions, providing a unified approach to geometric calculations.

This product's versatility shines in applications like reflections and rotations. By leveraging its properties, we can easily perform complex geometric operations, making it a cornerstone of geometric algebra's practical utility in various fields.

Geometric product of vectors

Definition and properties

• The geometric product of two vectors $a$ and $b$ is defined as $ab = a \cdot b + a \wedge b$
  • $a \cdot b$ is the scalar product (dot product)
  • $a \wedge b$ is the wedge product (outer product)
• The geometric product is associative, meaning $(ab)c = a(bc)$
• The geometric product is not commutative, meaning $ab \neq ba$ in general
• The square of a vector $a$ under the geometric product is a scalar equal to the squared magnitude of the vector: $aa = a \cdot a = |a|^2$

Scalar and bivector components

• The geometric product of a vector with itself is always a scalar
• The product of two distinct vectors generally consists of both scalar and bivector parts
  • The scalar part represents the projection of one vector onto the other, scaled by the magnitude of the other vector
  • The bivector part represents an oriented plane segment with magnitude equal to the area of the parallelogram formed by the two vectors

Scalar and bivector parts

Interpretation of scalar part

• The scalar part of the geometric product, $a \cdot b$, represents the projection of one vector onto the other, scaled by the magnitude of the other vector
  • It is a measure of the "overlap" or similarity between the two vectors
• The scalar product is symmetric, meaning $a \cdot b = b \cdot a$

Interpretation of bivector part

• The bivector part of the geometric product, $a \wedge b$, represents an oriented plane segment
  • The magnitude of the bivector is equal to the area of the parallelogram formed by the two vectors
  • The orientation is determined by the order of the vectors in the product
• The wedge product is antisymmetric, meaning $a \wedge b = -b \wedge a$

Special cases

• If the geometric product of two vectors is purely scalar ($a \wedge b = 0$), the vectors are collinear
• If the geometric product of two vectors is purely bivector ($a \cdot b = 0$), the vectors are perpendicular

Geometric product and vector angle

Relationship between scalar part and angle

• The scalar part of the geometric product is related to the angle $\theta$ between the vectors: $a \cdot b = |a||b| \cos(\theta)$
• When the vectors are normalized (unit vectors), the scalar part directly represents the cosine of the angle between them

Relationship between bivector part and angle

• The magnitude of the bivector part is related to the angle: $|a \wedge b| = |a||b| \sin(\theta)$
• The ratio of the bivector to the scalar part gives the tangent of the angle: $|a \wedge b| / (a \cdot b) = \tan(\theta)$

Encoding angle in the geometric product

• When the vectors are normalized (unit vectors), the geometric product directly encodes the angle: $ab = \cos(\theta) + \sin(\theta) B$
  • $B$ is the unit bivector representing the plane of the vectors
• This representation allows for easy extraction of the angle between vectors from their geometric product

Vector reflections and rotations using the geometric product

Reflections

• The reflection of a vector $a$ in a unit vector $n$ is given by $-nan$
  • This follows from the properties of the geometric product and the fact that $nn = 1$
• The reflection formula can be derived by decomposing $a$ into parts parallel and perpendicular to $n$, applying the properties of the geometric product, and simplifying

Rotations

• Rotations can be expressed as two successive reflections
  • If $n$ and $m$ are unit vectors, the expression $-m(nan)m$ rotates $a$ by twice the angle between $n$ and $m$ in the plane spanned by $n$ and $m$
• Rotors, which are even multivectors that encode rotations, can be constructed from the geometric product of two unit vectors as $R = nm$
  • The rotation of a vector $a$ is then given by $RaR^*$, where $R^*$ is the reverse of $R$
• Using rotors allows for compact representation and efficient computation of rotations in geometric algebra