# Questions tagged [geometric-algebras]

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

156 questions
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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>.
1answer
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### 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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### 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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### 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 ...
1answer
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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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### 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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### 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 ...
1answer
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### 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 ...
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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### 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 ...
1answer
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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 ...
1answer
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### 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 ...
1answer
230 views