Help with clifford algebra vector identity

[JBrandonS]

Homework Statement

This is question 1.1 from section 2-1 of New Foundations of Classical Mechanics:

Establish the following "vector identities":
[itex] (a\wedge b) \cdot (c \wedge d) = b\cdot ca \cdot d - b\cdot da \cdot c = b\cdot(c\wedge d)\cdot a[/itex]

Homework Equations

The Attempt at a Solution

My attempts at this solution make me believe that there is a typo in this problem. The quickest way is by using the third equation:

[itex]b\cdot(c\wedge d)\cdot a = b \cdot \frac{1}{2}(cd-dc)\cdot a = \frac{1}{2}(b\cdot cd \cdot a - b \cdot dc \cdot a)[/itex]

This is equal to half the second equation since [itex]a \cdot b = b \cdot a[/itex]. So am I doing something wrong?

 

[brombo]

GA BAC CAB Relations

The generalized BAC CAB relations are shown in the attached file. All the relations were generation using software for the symbolic manipulation of multivectors to be found at

https://github.com/brombo/GA

This repository also contains notes on geometric algebra based on Doran and Lasenby. The symbolic software (python modules using sympy) is described in great detail in the "LaTeX docs" directory. It is really easy to make mistakes when doing multivector manipulations by hand.