# Questions tagged [geometric-algebras]

Geometric algebras are Clifford algebras over the real numbers. They are applied in geometry and theoretical physics.

351 questions
Filter by
Sorted by
Tagged with
42 views

• 1,244
66 views

### Dot product of two exterior products and associativity of geometric product?

This is a quick and basic question. I looked online (Wikipedia articles, Wolfram, etc..., and poked inside of Hestenes and Snygg's books, but couldn't easily pull out an answer). I'm going to define ...
• 894
1 vote
39 views

### Help with Doran's book GA for Physicists [closed]

On page 213 equation (6.262) There is a formula which involves the Riemann tensor $a \cdot \mathbf R (b \land c) + c \cdot \mathbf R (a \land b) + b \cdot \mathbf R (c \land a) = 0$. Then Doran ...
84 views

### On the dot product of a vector with a bivector

I have this identity $$a\cdot (b \wedge c) = (a \cdot b)c - (a \cdot c)b$$ which I can prove pretty straight forward, using that $X\cdot Y = XY_\parallel$ and $X \wedge Y = XY_\perp$ (Where I break ...
56 views

### Deriving the Maximum Range of a Particle With A Constant Force Using Geometric Algebra

I am learning Geometric Algebra by reading New Foundations for Classical Mechanics, by David Hestenes. Chapter 3-2 studies the motion of a particle with constant gravitational force and on page 129 ...
67 views

### Problems in understanding exercise 2.2. from Chris Doran "Geometric Algebra for physicists"

I am currently trying to learn geometric algebra with Dorans book. I tried to do excercise 2.2., but I get stuck at the formulation of the exercise: By expanding the bivector $a \wedge b$ in terms of ...
• 43
83 views

### In geometric algebra, what is the dot product of a vector and a scalar? what is the wedge product of a vector and a scalar?

I am watching a series on geometric calculus by Alan Mcdonald and in the first episodes he states that for any vector u and multivector M: $uM = u \cdot M + u \land M$ This doesn't really take into ...
• 43
1 vote
49 views

### Trying to prove Euler's formula in 3D Geometric Algebra (Clifford algebra)

I'm a non-mathematician trying to learn geometric algebra by working my way (slowly) through "Imaginary Numbers are not Real" by Gull, Lasenby and Doran [Found. Phys. 23(9), 1175-1201 (1993)]...
• 177
93 views

### What is the relation of the metric matrix with the signature of a Geometric Algebra?

As far as I know, the metric matrix is used to measure the length of vectors regardless of the chosen basis used to represent them. I read in the book "Álgebra Geométrica e Aplicações" by ...
28 views

• 889
60 views

### Interior, cross and outer products between two multivectors?

For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of the interior (or scalar) product $\mathbf u\cdot \mathbf v$, the cross product $\mathbf u\times \mathbf v$ (if ...
• 593
32 views

### Is there a single term that generalizes the names of the individual vector and scalar types in a multivector?

In three dimensions, a multivector consists of a scalar, a vector, a bivector and a tri-vector. Is there a term that generalizes these names? For example, in an $n$-dimensional space, can I use the ...
• 593
224 views

### Fundamental complex number formula kept secret?

I recently started self-studying of complex analysis. I checked several popular online courses, and all of them, of course, start with the basic definitions: real and imaginary parts of a complex ...
• 1,120
1 vote
69 views

### Can the scalar product of the geometric algebra be defined without a determinant？

The scalar product, as defined in the book "Geometric-algebraic-for-computer-science" ,is directly based on this determinant. Is it possible for this definition to be independent of the ...
1 vote
73 views

### In geometric algebra, what is the grade of a k-vector in an n-dimensional space？ [closed]

It's a stupid question, but I am not figure it out. In the book I read, a k-vector in an n-dimensional space is defined as a linear combination of this space k-blade. I'm curious what is the grade of ...
120 views

### Is the inverse of a $k$-vector a $k$-vector?

In geometric (Clifford) algebra, both $k$-vectors and inhomogeneous multivectors may have inverses, which are unique (if they exist). I want to prove the following statement. Let $A = ⟨A⟩_k$ be a $k$-...
• 658
1 vote
56 views

• 13
55 views

### Do unit bivectors square to -1 in vector spaces of any dimension?

We know that in a 2D vector space (VS) the unit bivector squares to -1. My question is: Is the geometric product of a unit bivector with itself equal to -1 in any VS, independently of the VS dimension?...
• 147
1 vote
94 views

### Can a vector have a wedge product with a scalar? And a geometric product?

One of the necessary requirements of a vector space is to be endowed with the multiplication by a scalar. I am now interested in other operations, the wedge and the geometric product in geometric ...
• 147
37 views

• 4,764
1 vote
78 views

### Is Clifford product only for scalars already defined?

hi i'm a high school student interested in Clifford algebra. Since it is hard for me to understand several things thus i hope to know as follows: Is Clifford product only for scalars already defined? ...
130 views

### Intersection of *segments* in 3D using projective geometric algebra?

There is a relatively common technique to find the intersection of two segments in 3D, which can be found in page 304 of Graphics Gems Note that a segment is a compact subset of a line. The technique ...
• 3,439
1 vote
139 views

### How can one generalize the split-complex algebra to a coordinate space with an arbitrary number of asymptotes?

Given that split-complex numbers generates 2D coordinates with 4 asymptotes (when multiplying) that look as follows: The arrows along the hyperbola indicate a positive direction for boosting points ...
36 views

### Orthogonal transformations and reflections

In the context of geometric algebra, why is it that any orthogonal transformation can be expressed as successive reflections? (Both, a mathematical and an intuitive explanation would be great)
114 views

### Representation of angular momentum as bivector

I read about derivation of angular velocity and angular momentum here - https://peeterjoot.com/archives/geometric-algebra/angular_acc.pdf Peeter Joot wrote: "writing the angular momentum as ...&...
• 889
1 vote
Given $p_1,p_2,p_3 \in S^2 = \{x \in \mathbb{R}^3 \mid \lVert x \rVert = 1\}$ all distinct, I wish to prove that \begin{align*} J &:=(p_1p_2 - 1)^2 + (p_1p_3 - 1)^2 + (p_2p_3 - 1)^2\\ &\qquad -...