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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.

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Write down the eqn. of the line that passes through the points $(4, -2, 1)$ and $(6, 0, 3)$ in all three forms.

I'm stuck because I can't find the parallel vector. Do you know how to find it? I'd just assumed that vector r0 = <4, -2, 1>.
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72 views

The commutator product as a replacement for the cross and wedge product in geometric algebra?

From an axiomatic approach to geometric algebra, the wedge product of two vectors $a$ and $b$ is typcially defined as the antisymmetric $a \wedge b = \frac{1}{2}(ab - ba)$, where $ab$ is the geometric ...
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Inner product structure on geometric algebra?

I understand that geometric algebra equips itself with the contraction operators $\rfloor$ and $\lfloor$. While these are awesome when one wishes to project a subspace onto another, it is not an inner ...
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25 views

Converting $p(\theta,t)=e^{a\wedge c\theta}(a+be^{a\wedge bt})e^{-a\wedge c\theta}$ from geometric to vector algebra

Let $a,b,c\in\mathbb{R}^3$ with $a\wedge b\wedge c\ne 0$ and $a^2>b^2$. What is the form of this equation $p(\theta,t)=e^{a\wedge c\theta}(a+be^{a\wedge bt})e^{-a\wedge c\theta}$ in standard ...
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137 views

Intuition for geometric product being dot + wedge product

While I feel quite comfortable with the meaning of the dot and exterior products separately (parallelity and perpendicularity), I struggle to find meaning in the geometric product as the combination ...
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Write a bivector as the exterior product of two vectors

The Wikipedia article https://en.wikipedia.org/wiki/Bivector#Simple_bivectors states that "A bivector that can be written as the exterior product of two vectors is simple. In two and three ...
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Character space of commutative c*-algebra

I would like to know if there exists a characterization for a commutative c*-algebras with sigma compact character space
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47 views

Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the Clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
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How to compute if a multivector inverse exists in Clifford Algebra

Suppose we have a 4 dimension positive signature clifford algebra. In Calculating the inverse of a multivector and Inverse of a general nonfactorizable multivector, the inverse of a multivector is ...
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61 views

Given $\textbf{a}\perp\textbf{b}$, $\textbf{a}\cdot(\textbf{a}\wedge\textbf{b})=\mid \textbf{a}\mid^{2} \textbf{b}$ (geometric algebra)

I want to prove, given $\textbf{a}\perp\textbf{b}$, $$\textbf{a}\cdot(\textbf{a}\wedge\textbf{b})=\mid \textbf{a}\mid^{2} \textbf{b}$$ I realize this is just a matter of $$\textbf{a}\cdot(\textbf{a}\...
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81 views

Deriving reduction formula in Geometric Algebra

I am trying to learn Geometric Algebra by going through the book "New Foundations for Classical Mechanics" by David Hestenes. I was reading the part about reduction formula (shown below) but couldn't ...
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How does one represent generalized polynomials in Conformal Geometric Algebra C(4,1)

I am interesting in representing arbitrary curves using conformal geometric algebra. I have a special interest in spatial loops. Also, I would like to represent characteristic polynomials of matrices ...
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1answer
89 views

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 ...
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Intuition for orientation of tri-vectors in geometric algebra

I am learning geometric algebra from the MacDonald textbook and it states that the outer product is associative. Letting $\bf{u}$, $\bf{v}$, and $\bf{w}$ be vectors $$\bf{u} \wedge \bf{v} \wedge \bf{...
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1answer
84 views

Visualizing the area described by the dot product?

Since the dot product of two vectors is an area (if your vectors have units of meters, then the dot product would be in m$^2$), I was wondering if there is a good way to visualize that area. The ...
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40 views

If $A^iB_i$ is called a contraction, what is $A^{ij}B_{ij}$ called?

I have a tensor $A^{ijk}$ and a tensor $B_{ijk}$, and I'd like to contract all the indices between them, resulting in the scalar $k$: $$k = A^{ijk}B_{ijk}$$ Is there a name for this sort of "multi-...
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what is the geometry picture of Riemann tensor identity $R(X\wedge Y,V\wedge W) = R(V\wedge W, X\wedge Y)$

For symmetries in Riemann tensor of $R(X,Y)V:=\nabla_X\nabla_YV-\nabla_Y\nabla_XV-\nabla_{[X,Y]}V$, there are excellent explanations on the intuition behind it like this relevant question. If we think ...
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Limits of overdot notation in geometric algebra

In geometric calculus the over dot notation is used to denote the proper way to do the vector differentiation of a multivector product - $$ \nabla (AB) = (\nabla A)B + (\dot{\nabla}A)\dot{B} $$ The ...
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2answers
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Formula (1.18) page 43 in Hestenes book “New foundations for Classical mechanics”

The formula is $(A_r\land b)\cdot C_s=A_r\cdot (b\cdot C_s)$, where $0<r<s$. Hestenes suggests to expand $(A_rb)C_s=A_r(bC_s)$ and extract the $s-r-1$-vector part. But this method requires one ...
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78 views

geometric algebra expression for a linear transformation defined on basis vectors

I have a transformation $T(u)$ between 3D vector subspaces of a 4D vector space, $A \rightarrow B$, defined by a mapping of their basis vectors: $A:\{a_1=e_1\,,\,a_2=e_2\,,\,a_3=e_4\}$ $B:\{b_1=e_2\...
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XKCD #2028 explenation: what is the diference between algebreic geometry and geometric algebra? [closed]

The 3rd of august XKCD was all about complex numbers and how mathematicians are too cool for regular vectors. the linked title was: I'm trying to prove that mathematics forms a meta-abelian group, ...
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81 views

Reflection in Geometric Algebra

When I reflect $v$ through the hyperplane orthogonal to $v-w$, where $w = f(v)$ and $f$ is an isometry on $\mathbb{R}^n$ that preserves the origin. I believe I am supposed to get $w$, however whenever ...
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164 views

Wedge Product on $C\ell^+(0,3,0)$ Relationship to Quaternion Cross Product

The even Clifford sub-algebra $C\ell^+(0,3,0)$ is isomorphic to the quaternion algebra. The mapping between terms is $e_0 \mapsto 1$, $e_{23} \mapsto i$, $e_{31} \mapsto j$, $e_{12} \mapsto k$. In ...
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Transformation of metric tensor in clifford algebra for nonorthogonal transformations

I study physics so apologies for any nonstandard notation/terminology. In geometric (aka spacetime) algebra one speaks of basis $\gamma^{\mu}$ ( possibly represented by matrices) as transforming in a ...
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2answers
126 views

The Fundamental Theorem of Geometric Calculus in a lorentzian manifold

I am trying to understand geometric calculus and apply it to physics. In this sense, I was reading Alan Macdonald's book "Vector and Geometric Calculus", and stumbled upon the Fundamental Theorem of ...
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1answer
71 views

Is there a multivector that's a non-trivial cube root of $1$?

This answer (or this site, in case the answer gets deleted) defines a certain 3-dimensional Real algebra by declaring that $j^3={^-}1$, and that $1,j,j^2$ are linearly independent. (By a simple ...
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How to decompose a bivector into a sum of _orthogonal_ blades?

In Geometric Algebra, any bivector $B\in\Lambda^2\mathbb R^n$ is a sum of blades: $$B = B_1 + B_2 + \cdots$$ $$= \vec v_1\wedge\vec w_1 + \vec v_2\wedge\vec w_2 + \cdots$$ Each blade's component ...
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Householder reflection in geometric algebra is not working for me

A Householder reflection of a vector $v$ along a direction $n$ is given by the formula \begin{eqnarray} v' = v - 2 \frac{n \cdot v}{|n|^2} n. \end{eqnarray} If $n$ is unitary then $v'=v-2 (n \cdot ...
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Geometric Calculus, Clifford Algebra, and Calculus of Variations

It has always bothered me that I was told in my Calculus of Variations class that it's only possible to minimize a function with respect to one objective. Obviously sometimes it is possible to ...
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For two unit non-oriented bivectors $A,B\in \mathbb{R}P^2\subset \Lambda^2\mathbb{R}^3$ is the mapping $\phi:(A,B)\rightarrow AB$ bijective?

For two non-oriented unit bivectors $A,B\in \mathbb{R}P^2\subset \Lambda^2\mathbb{R}^3$ is the mapping $\phi:\mathbb{R}P^2\times \mathbb{R}P^2/\mathbf{D} \rightarrow S^3$, where $\mathbf{D}$ is the ...
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140 views

What is exponential of a blade?

This answer to another question discusses using geometric algebra to find a rotation of an arbitrary angle between two vectors. It involves constructing a versor $R$ from a normalized blade $\hat{B}$ ...
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Reverse operation on Quaternions

I'm currently studying Clifford algebras and I came across a concept called reverse. It has the following properties: $(AB)^† = B^†A^†$ for all $A$ and $B$. $v^† = v$ for all vectors $v$. I was ...
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1answer
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A confusing formula in Clifford algebra

I am reading a book named "An Introduction to Clifford Algebras and Spinors" by J. Vaz Jr. and R. da Rocha. In page 78, I met a confusing formula (3.89), written as: $$\gamma(\mathbf{v})\gamma(\...
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Conditions for a C*-algebra to be Abelian

Please help me with this problem. I don't know how to get into proof. Let $A$ be a $C^*$-Algebra. Show that if the condition $0 \leq x \leq y,~~ x,y\in A,$ implies $x^2 \leq y^2,$ then $A$ is abelian....
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103 views

A new type of curvature multivector for surfaces?

A surface, parametrized by $U$ and $V$, has a tangent bivector given by the wedge product $$I = \vec x_U\wedge\vec x_V$$ where subscripts represent partial derivatives. The First Fundamental Form ...
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132 views

Geometric Algebra Rejection, Projection and reflection rotation, confused on how end result is actually calculated

I recently found a video that claimed to give intuition to Quaternions. And to my suprise it nearly did, but I have a few large hang ups on the lack of definitions of certain operations, it also ...
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1answer
149 views

General expression for geometric product of blades in terms of scalar and exterior products

I am trying to find a general expression for the geometric product of two blades in terms of the scalar and exterior products of vectors. Some preamble to be clear on conventions: In a geometric ...
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1answer
76 views

Exterior product between dual vectors in homogeneous coordinates

In 3D-projective geometry the homogeneous coordinates of the line connecting two points with homogeneous coordinates $(x_0 : x_1 : x_2 : x_3)$ and $(y_0 : y_1 : y_2 : y_3)$ can be calculated as a kind ...
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3answers
490 views

Generalizing the dot product to multivectors

I am studying the book Linear and Geometric Algebra (Macdonald), and I've been stuck on a couple related, seemingly-elementary problems for a couple of days. 5.3.4. Suppose that $\mathbf{a} \bot \...
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2answers
564 views

Geometric algebra: Rotation of a rotor

In short my question is: Why is the rotation of a rotor in geometric algebra implemented by a single-sided rotation? To elaborate: In geometric algebra, rotating an object is done by multiplying it ...
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2answers
86 views

Homomorphism Between a Geometric Algebra and its Field of Scalars?

Given a geometric algebra defined over a real vector space, is is possible to construct a homomorphism between the elements of the geometric algebra and the reals? I was pondering an example of this: ...
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1answer
120 views

Geometric algebra and simple geometric operations

I'm not an expert on the subject, but If it is worth I'd like to start on getting some grasp on the subject. Assuming geometric algebra framework, Is there somewhere a list of formulas where for ...
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1answer
105 views

Equivalence of Left and Right Inverse in Geometric Algebra

I am asked by StackExchange to clarify if this is a duplicate question. This is not a duplicate of the question about matrix multiplication. This question is not (on its surface) a question about ...
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1answer
130 views

Solving for an unknown rotor in a real geometric algebra

So, I'm working in a real geometric / Clifford algebra generated by a set of 3 orthonormal vectors $\text{e}_1,\text{e}_2,\text{e}_3$ all with positive squares. I came to this rotor equation and I'm ...
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1answer
316 views

Does the cross product of $\mathbb{R}^3$ produce a 1-vector or a 2-vector?

I think that, in $\mathbb{R}^3$, 1-vectors and 2-vectors have 3 components. Here, also, the cross product $\vec a \times \vec b$ and the exterior product $\vec a \wedge \vec b$ of vectors $\vec a$ and ...
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1answer
647 views

Is Hestenes's Geometric Algebra widely accepted?

I would like to write a paper on the fundamentals of Continuum Mechanics using the Geometric Algebra approach popularized by David Hestenes. Is Hestenes's Geometric Algebra a wide accepted theory? I'...
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780 views

Are Clifford and exterior algebras isomorphic as “wedge product algebras”?

$\newcommand{\Cl}{\mathscr{Cl}(V)}$$\newcommand{\ext}{\Lambda(V)}$Let $V$ be a finite dimensional vector space over a field with characteristic not equal to two. Assume we have made a choice of ...
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Is there a relationship between the trace and the Clifford/geometric product?

In what follows, let $V=\mathbb{R}^n$ (although the following probably applies also to a larger number of finite-dimensional spaces). We assume throughout that we have made a choice for an inner ...
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183 views

Is this interpretation of differential forms within the context of geometric algebra correct?

Note: Geometric algebra here refers to the Clifford algebra of Euclidean space, and thus is distinct from algebraic geometry, although both are related to differential forms (to the best of my ...
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94 views

quaternion composition in geometric algebra

I was studying the book Linear And Geometric Algebra and I've been stuck at this problem for couple of days. Assume $\{e_1,e_2,e_3\}$ is an orthonormal basis in $\mathbb{R}^3$, and two quaternions $...