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Questions tagged [geometric-algebras]

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

Geometric algebras are Clifford algebras over the real numbers. They can be used as a tool to study vector algebra, and they can be applied to problems in geometry and theoretical physics.

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geometric algebra: dot products of vectors with vectors vs bivectors

The book "Matrix Gateway to Geometric Algebra" by Garret Sobczyk gives the following identity: $a\cdot(b \wedge c)=(a\cdot b)c - (a\cdot c)b$ Using the identities $a\cdot b= (ab+ba)/2$ $a\wedge b=(ab-ba)/2$ to expand the proposed identity, including…
Gary Lomp
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Show a rotation is given by $v=e^{-i\theta }ue^{i\theta }$

I am currently reading Alan Macdonald's book on Linear and Geometric Algebra. I can't seem to figure out a line: Why is line 4 true? The 3 previous lines make sense. Thanks for any insights you may provide!
Fernando Garcia Cortez
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What is the expression to evaluate the geometric product of an r-blade with an s-blade?

I am learning geometric algebra from David Hestene's New foundations to classical mechanics, in it the geometric product of two vector is defined the following way: $$ ab = a \cdot b + a \wedge b$$ And, the product of a vector with an $r$ blade is…
tryst with freedom
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Identity element in geometric algebra and computation method for finding the inverse of a multivector

In the book New foundations for classical mechanics by David Hestenes, the following equation is supposed to be the defining equation multiplicative inverse for a multivector: $$A^{-1} A = 1$$ Is the $1$ on the RHS a scalar? 1, 2
tryst with freedom
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Exponential Sine Identity in Geometric Algebra

Given $$e^{\textbf{i}\theta}=cos(\theta) + \textbf{i}sin(\theta) \ \ \ \ (1)$$ and $$e^{-\textbf{i}\theta}=cos(\theta) - \textbf{i}sin(\theta) \ \ \ \ (2)$$ To find $sin(\theta)$ first you should…
roshoka
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Projections in Geometric Algebra

Given a geometric algebra $G$. Let $I, A,B$ be an $n,r$ and $s$ blade, respectively. Let $P(*)=(*\rfloor I) I^{-1}$ be the projection operator onto $I$. Suppose that $P(A)=A$. The problem is to show that $P(AB)=AP(B)$. I'm able to show this…
Jason Rodriques
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Question about domains of Geometric (Clifford) Calculus functions

For those of you familiar with Geometric Calculus, and in particular, the book Vector and Geometric Calculus by Alan MacDonald, maybe you can explain something to me. On pg 16, MacDonald defines the notion of limit (non-rigorously, but that's fine…
user137731
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Proving det(A) property with outermorphism definition

Let A be a square matrix. Prove that exchanging two columns of A changes the sign of $\det(A)$. **Note: I'm pretty sure this is supposed to be "adjacent columns" Source: "Linear and Geometric Algebra" - Alan MacDonald The definition of $\det(A)$…
user137731
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Geometric Algebra Question

Show that in 3D any pair of bivectors A and B have a common factor u such that A = au and B = bu. (a, b, u vectors -- au and bu are the geometric product) The only thing I can think of is to try putting everything in component form, but I keep…
user137731
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wedge product between nonorthogonal basis and its reciprocal basis in geometric algebra

in the context of geometric algebra, what's the wedge product between basis and reciprocal basis? say, if {$e_i$} is a set of basis that are not necessarily orthogonal, and {$e^i$} is the corresponding reciprocal basis, we have $$e_i \cdot e^j =…
ahala
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inner product of trivector and bivector in geometric algebra

Hestenes's "New Foundations for Classical Mechanics" book (page 47, 1.1c) sets a problem to show: $\begin{aligned}\left( \mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c} \right) \cdot B=\mathbf{a} \left( \left( \mathbf{b} \wedge \mathbf{c} \right)…
Peeter Joot
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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 follows from $\partial \wedge x = 0,…
Rodrigo
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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: How to derive $(a \cdot b)^2 - (a \land b)^2 =…
Mike_bb
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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 determinant? I would like to understand its motives better (of…
Takayama_Maria
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In geometric algebra, what is the grade of a k-vector in an n-dimensional space？

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 the k- vector, is it also k?
Takayama_Maria
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