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I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby.
They claim in chapter 4 that the geometric product [itex]ab[/itex] between two vectors [itex]a[/itex] and [itex]b[/itex] is defined according to the axioms
i) associativity: [itex](ab)c = a(bc) = abc[/itex]
ii) distributive over addition: [itex]a(b+c) = ab+ac[/itex]
iii) The square of any vector is a real scalar
Then they claim that the inner and outer product are defined as
[itex] a \cdot b = \frac{1}{2} (ab+ba) [/itex]
[itex] a \wedge b = \frac{1}{2} (ab-ba) [/itex]
so that
[itex] a b = a \cdot b + a \wedge b [/itex]
My problem is that if you are given two vectors, say
[itex] a = 1e_1 + 3 e_2 - 2e_3 [/itex]
[itex] b = 5e_1 -2 e_2 + 1e_3 [/itex]
How to actually compute [itex]ab[/itex]?
I mean, you then have to specify how either [itex] a \cdot b [/itex] and [itex] a \wedge b [/itex] works, or how (in detail) [itex] a b [/itex] are to be performed.
This is in my view, circular.
From another point of view, say that you start backwards by defining the inner and outer product, and then define the geometric product as [itex] a b = a \cdot b + a \wedge b [/itex]
Then how to show that the geometric product is associative? The usual definition of the outer product is associative, but the usual definition of the inner (dot) product is NOT associative. So how to show that the geometric product is associative if you take the inner and outer product as starting point for the geometric product?
Thank you very much in advance for any kind of feedback
They claim in chapter 4 that the geometric product [itex]ab[/itex] between two vectors [itex]a[/itex] and [itex]b[/itex] is defined according to the axioms
i) associativity: [itex](ab)c = a(bc) = abc[/itex]
ii) distributive over addition: [itex]a(b+c) = ab+ac[/itex]
iii) The square of any vector is a real scalar
Then they claim that the inner and outer product are defined as
[itex] a \cdot b = \frac{1}{2} (ab+ba) [/itex]
[itex] a \wedge b = \frac{1}{2} (ab-ba) [/itex]
so that
[itex] a b = a \cdot b + a \wedge b [/itex]
My problem is that if you are given two vectors, say
[itex] a = 1e_1 + 3 e_2 - 2e_3 [/itex]
[itex] b = 5e_1 -2 e_2 + 1e_3 [/itex]
How to actually compute [itex]ab[/itex]?
I mean, you then have to specify how either [itex] a \cdot b [/itex] and [itex] a \wedge b [/itex] works, or how (in detail) [itex] a b [/itex] are to be performed.
This is in my view, circular.
From another point of view, say that you start backwards by defining the inner and outer product, and then define the geometric product as [itex] a b = a \cdot b + a \wedge b [/itex]
Then how to show that the geometric product is associative? The usual definition of the outer product is associative, but the usual definition of the inner (dot) product is NOT associative. So how to show that the geometric product is associative if you take the inner and outer product as starting point for the geometric product?
Thank you very much in advance for any kind of feedback