Skip to main content

Questions tagged [geometric-algebras]

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

Filter by
Sorted by
Tagged with
0 votes
1 answer
42 views

How does the fact that $\mathbb{G}^n$ is $2^n$ dimensional help show that the $\mathbb{G}^n$ norm satisfies the triangle inequality?

I'm home-studying geometric algebra from Alan Macdonald's "Linear and Geometric Algebra" textbook. Problem 6.2.1 reads Show that the $\mathbb{G}^n$ norm satisfies the properties of the $\...
KCE's user avatar
  • 235
3 votes
0 answers
43 views

Is there a geometric interpretation of symmetric powers of vectors?

Let $V$ be a finite-dimensional vector space, over the real numbers to fix ideas. If we pick an inner product then the classical image of a vector in $V$ is a directed arrow. If $v_1, \ldots, v_k$ are ...
Gunnar Þór Magnússon's user avatar
0 votes
0 answers
36 views

What general concept, other than oriented volume, embodies all even (odd) permutations of $n$ elements?

Given an ordered orthogonal basis $v_1...v_k$ of a $k$-dimensional vector space, we can construct a $k$-vector in the geometric algebra over that space, simply by multiplying the basis vectors in the ...
Adam Herbst's user avatar
0 votes
1 answer
57 views

Help with proof of Curl Double Product identity using Geometric Algebra. Most things seem to fall in place, but having a few issues.

So I'm pretty new to GA/Clifford Algebras, but it's been fairly interesting so far. I figured I'd try to prove some basic vector calculus identities with it, just to help me get my bearings. I decided ...
Copywright's user avatar
0 votes
1 answer
45 views

Meaning of adding planes represented in the same basis

According to "Clifford algebra of points, lines and planes" by Selig, a plane can be described by a unit normal vector $\boldsymbol{n}$ and a perpendicular distance $d=\boldsymbol{r} \cdot \...
CroCo's user avatar
  • 1,244
0 votes
1 answer
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 ...
Nate's user avatar
  • 894
1 vote
0 answers
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 ...
alexanderyaacov's user avatar
2 votes
1 answer
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 ...
Ilikemath's user avatar
0 votes
1 answer
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 ...
Cereza's user avatar
  • 1
4 votes
1 answer
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 ...
JLJ's user avatar
  • 43
3 votes
2 answers
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 ...
Minimo's user avatar
  • 43
1 vote
1 answer
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)]...
MartinC's user avatar
  • 177
0 votes
2 answers
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 ...
ivansnpmaster's user avatar
0 votes
1 answer
28 views

If $A = P(A) = \langle A\rangle_r$, why is $\dot \partial (\dot x \cdot A) = A\cdot \partial x$ (Geometric algebra)

Let $x$ be a vector of an $n$-dimensional subspace $\mathcal A_n$ with $P$ the projection operator onto this subspace. Then, let $A= P(A) = \langle A\rangle_r$ be an $r$-vector. Then, I know that $\...
Rodrigo's user avatar
  • 7,716
1 vote
2 answers
60 views

Prove that $\partial x = \partial \cdot x = n$ (Geometric algebra)

I’m trying to understand equation (2-1.34) on page 51 of Hestenes and Sobczyk’s “Clifford Algebra to Geometric Calculus”. $\partial x = \partial \cdot x = n \tag{1.34}$ According to the book, this ...
Rodrigo's user avatar
  • 7,716
0 votes
1 answer
58 views

Show that $\partial x^2 = 2P(x)$ and $\partial \wedge x = 0$ (Geometric algebra)

Show that $\partial x^2 = 2P(x)$ and $\partial \wedge x = 0$ (Geometric algebra)I’m trying to show equations (2-1.32) and (2-1.33) on page 51 of 20 of Hestenes and Sobczyk’s “Clifford Algebra to ...
Rodrigo's user avatar
  • 7,716
0 votes
2 answers
78 views

Why is the inner product of two simple vectors simple? (Geometric algebra)

I’m trying to understand the reason for the assertion on page 20 of Hestenes and Sobczyk’s “Clifford Algebra to Geometric Calculus” that If $B$ is a simple $s$-vector, then $B\cdot A$ [where $A$ is a ...
Rodrigo's user avatar
  • 7,716
1 vote
1 answer
99 views

How to obtain $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$ and why $(a \land b)=|a||b||\sin(\phi)|$ if $(a \land b)^2=-|a|^2|b|^2\sin^2(\phi)$? (GA)

I read in Wiki (https://en.wikipedia.org/wiki/Bivector) that antisymmetric part of geometric product can be represent as $(a \land b)$ and $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$ I have 2 questions: ...
Mike_bb's user avatar
  • 889
1 vote
0 answers
82 views

Geometric Algebra: No other point groups besides $H_p$ and $C_p$ in two dimensions

I have been reading the paper "Point Groups and Space Groups in Geometric Algebra" by David Hestenes as part of my introduction seminar to GA. On page 5 the remark is made, that to prove ...
user1292126's user avatar
0 votes
0 answers
36 views

How to get scalar product from geometric product in Geometric Algebra?

I read that geometric product $ab$ can be decomposed into 2 parts (symmetric and antisymmetric). But I can't understand why symmetric part is a scalar product. I mean following symmetric part: $\frac{...
Mike_bb's user avatar
  • 889
0 votes
1 answer
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 ...
HelloGoodbye's user avatar
0 votes
1 answer
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 ...
HelloGoodbye's user avatar
-4 votes
1 answer
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 ...
Alex C's user avatar
  • 1,120
1 vote
1 answer
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 ...
Takayama_Maria's user avatar
1 vote
1 answer
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 ...
Takayama_Maria's user avatar
3 votes
1 answer
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 $k$-...
Jollywatt's user avatar
  • 658
1 vote
1 answer
56 views

Viewing vectors in a Clifford algebra as reflections

Let $Cl(s,t)$ be the Clifford algebra over $\mathbb{R}^{s,t}$ where $(s,t)$ is the signature of the bilinear form $\eta$. Let $Pin(s,t)$ be the associated pin group and define $$R: Pin(s,t) \times \...
CBBAM's user avatar
  • 6,255
2 votes
0 answers
110 views

What is the intuition behind the regressive product and its axioms?

I'm self-studying Grassmann algebra and I find it hard to visualize the regressive product, specifically the magnitude and orientation of the outcome. My main soure is John Brown's Grassman Algebra ...
Ilikemath's user avatar
0 votes
1 answer
50 views

Computing $x\cdot e_n \cdot x \cdot e_n$ in a Clifford algebra over $\mathbb{C}^d$

I am closely following Hamilton's Mathematical Gauge Theory. Let $V$ be a vector space, $Q$ a bilinear form on $V$, and $CL(V,Q)$ the corresponding Clifford algebra. We can construct the Clifford ...
CBBAM's user avatar
  • 6,255
1 vote
0 answers
83 views

What is a Geometric Interpretation of the Product of two Blades?

I am hoping for a meaningful interpretation for the geometric product of two blades of not-necessarily the same grade. I understand that blades can be expressed as a sum of basis k-vectors, and then I ...
NicNic8's user avatar
  • 7,032
0 votes
0 answers
81 views

How to derive the Log and Exp functions of a Geometric Algebra Rotor for Spherical Interpolation (Slerp) in 4D

I am trying to create a Slerp function between two 4D Rotors Rotor Slerp(Rotor a, Rotor b, t) { ... } I have been told you can do an interpolation using Log and ...
Joe's user avatar
  • 1
1 vote
2 answers
90 views

Reference: Is a simple k-vector an equivalence class?

My understanding of a simple k-vector is that it is the wedge product of k vectors. Also, two simple k-vectors are the same, when their magnitude, attitude and orientation match. Now my question is, ...
dontknow3's user avatar
3 votes
0 answers
132 views

Cartan's magic formula and geometric product

Cartan's formula say's $L_X = d \circ i_X + i_X \circ d$ while in geometric algebra, we have $ab = a \cdot b + a \wedge b$. These formula's seem similar to me. Especially, because the first is about k-...
Jelle Bloemendaal's user avatar
1 vote
2 answers
82 views

Show that $\tfrac{1}{2}(b B\cdot (c \wedge a) - a B\cdot (c \wedge b)) = (c\cdot b) a \cdot B $ in geometric algebra

I would like to show that $$\frac{1}{2}\left( b B\cdot (c \wedge a) - a B\cdot (c \wedge b)\right) = (c\cdot b) a \cdot B $$ where B is a bivector, and the others are vectors. I've tried to coerce ...
foghorn's user avatar
  • 209
0 votes
1 answer
124 views

Intuition for why a 90 degree rotation of a vector about an arbitrary axis can be expressed as 3 90 degree rotations of the vector's projections.

Given a unit vector $\hat{u}$ and a vector $\vec{v}$ perpendicular to $\hat{u}$, we can rotate $\vec{v}$ by 90 degrees around $\hat{u}$ with the cross product $\hat{u} \times \vec{v}$. Since the cross ...
someguy67's user avatar
  • 579
1 vote
0 answers
40 views

Norm of a Multivector in $\wedge \mathbb{R}^3$ for calculating the arrea of a polygon.

I am writing some code to explore some interesting things in Geometric Algebra. The general element of my code is multivector $\wedge \mathbb{R}^3$ that forms an 8-dimensional block vector with ...
John Alexiou's user avatar
  • 14.2k
0 votes
1 answer
71 views

On vector multiplication [closed]

In this video (check it out, it's worth it), F. Holmér nicely derives the dot and cross product (with some insights into quaternions, the wedge product and much more), just by using ordinary ...
ric.san's user avatar
  • 141
0 votes
0 answers
77 views

Do the Clifford Algebra products $e_1e_0$, $e_1e_{-1}$ and $e_0e_{-1}$ anticommute?

In a Clifford Algebra $\mathbb{CL}_{(1,1,1)}$ we have the following relations: $e_1^2=1$ $e_{-1}^2=-1$ $e_0^2=0$ Question: Do all the products $e_ie_j=-e_je_i$ anticommute? If so, why? $\\ % blank ...
tutizeri's user avatar
  • 166
2 votes
1 answer
81 views

Geometric Algebra: show $ A_r \cdot B_s = (-1)^{r(s-1)} B_s \cdot A_r$ and $A_r\wedge B_s = (-1)^{rs} B_s \wedge A_r$ from Hestenes and Sobczyk's book

I'm making a go at self-study from Hestenes and Sobczyk's book Clifford Algebra to Geometric Calculus. I'm stuck on the simple formulas in the first section for reversing the order for the inner and ...
Kyle Taljan's user avatar
1 vote
1 answer
229 views

Quaternion Integration with Angular Acceleration

I'm looking to integrate angular velocity and acceleration into a quaternion. After researching it, I came across this site, which shows that integrating angular velocity alone gives the following: $$...
Jacob FW's user avatar
0 votes
0 answers
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?...
TrentKent6's user avatar
1 vote
2 answers
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 ...
TrentKent6's user avatar
2 votes
1 answer
37 views

Proof of bivector multiplication with reciprocal frame vector from Doran, Lasenby?

Doran and Lasenby (Geometric Algebra for Physicists) introduce the reciprocal frame vector and make the below assertion about multiplication with arbitrary bivectors (page 102, eq 4.104): $e_i e^i \...
chockeyblocky's user avatar
2 votes
3 answers
177 views

A proof for bivector decomposition

Let $V$ be an $n$-dimensional vector space over some field $\mathbb F$. I'm interested in the following result: For every bivector $\alpha\in\bigwedge^2 V$, there exists a linearly independent set $S=...
ViHdzP's user avatar
  • 4,764
1 vote
1 answer
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? ...
mathgoblins's user avatar
2 votes
0 answers
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 ...
Makogan's user avatar
  • 3,439
1 vote
1 answer
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 ...
Rehno Lindeque's user avatar
0 votes
0 answers
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)
TheoryWiz's user avatar
0 votes
1 answer
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 ...&...
Mike_bb's user avatar
  • 889
1 vote
2 answers
177 views

Inequality involving dot products of sphere points

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 -...
hhliu's user avatar
  • 15

1
2 3 4 5
8