# 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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### 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?...
1 vote
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### 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 ...
1 vote
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### 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? ...
43 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 ...
1 vote
104 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 ...
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### 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)
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### 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 ...&...
1 vote
159 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 -...
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### Why does the wedge product produce a scalar and not a graded element?

Reading the definition of the geometric product we get that it satisfies: $$a_1 \wedge \cdots \wedge a_r = \frac{1}{r!} \sum_{\sigma \in G_r} \mathrm{sgn}(\sigma)a_{\sigma(1)}\cdots a_{\sigma(r)}$$ ...
240 views

### Solving for the rotor in quaternion rotation

If we have 2 vectors $v_1,v_2$ which have been rotated into $v'_1,v'_2$ by the following operations: $v'_1 = e^{θ\hat{n}/2}v_1e^{-θ\hat{n}/2}$ $v'_2 = e^{θ\hat{n}/2}v_2e^{-θ\hat{n}/2}$ Where $\hat{n}$ ...
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On page 40 of Geometric Algebra for Physicists, Doran states in equation 2.98 that for a unit vector $n$ and an arbitrary vector $a$ with $a_\perp = n n\wedge a$, that $$n\cdot a_\perp=\langle n n \... 1 vote 2 answers 37 views ### How do units work for multivectors? (Clifford/Geometric algebra) [closed] Suppose we work with \mathcal{G}(2). If the units of e_{1} and e_{2} are of length, it makes sense that the units of e_{12} would be of area. But then what are the units of scalars? And what ... 0 votes 1 answer 28 views ### Outer product for r vectors in terms of the geometric product I found a definition in (An introduction to geometric algebra and calculus by Alan Bromborsky) which say let a_1,a_2,...,a_r be vectors in \mathbb{R}^n and let \varepsilon_{1...r}^{i_1...i_r} be ... 3 votes 1 answer 75 views ### For what types of objects is the outer product defined in geometric algebra? I just started to learn geometric algebra from the "Geometric Algebra for Physicists" book. Authors first give definition of outer product of two vectors a \wedge b. Then they give ... 0 votes 0 answers 42 views ### Solving a equation with a 2-form/bivector differential equation I was thinking about a problem and a bivector differential equation appeared, not sure how to start solving the issue. I want to find parameterization of the n-sphere, we know the implicit equation of ... 0 votes 1 answer 70 views ### Taking the geometric derivative of e^{\frac{1}{2}\left[\gamma_0 \vec{k} \vec{x} \gamma^0 - \vec{x} \gamma^0 \gamma_0 \vec{k}\right]} As in the title: I'm looking to take the geometric derivative with respect to \vec{x} of the exponential of the commutator of \gamma_0 \vec{k} and \vec{x} \gamma^0:$$e^{\frac{1}{2}\left[\...
The wedge product of two vectors $\vec{v}, \vec{w}\in\mathbb{R}^{n}$ can be defined as an anti-symmetrized tensor product. In three dimensions, there is a correspondence between the wedge product of ...