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 get the result the shown in the book. Can someone show me how iterating (1.15) gives the reduction formula?
notation in the book:
- dot (.) is inner product
- circumflex (^) is outer product
- ab mean geometric product of a and b
- inner and outer product have precedence over geometric product unless indicated by parentheses
I can easily apply (1.15) for the first iteration and get
$a \cdot (a_1 \wedge a_2 \wedge \cdots \wedge a_r) = (a \cdot a_1) (a_2 \wedge \cdots \wedge a_r) - a_1 \wedge (a \cdot (a_2 \wedge a_3 \cdots \wedge a_r))$
I can see that I should apply (1.15) to the term
$- a_1 \wedge (a \cdot (a_2 \wedge a_3 \cdots \wedge a_r))$
but there is a $ - a_1 \wedge $ in the term which will get inherited if I apply (1.15) directly, what should I do to get rid of that?