I am reading this PDF and have a question regarding an example which is given page $9$ with regard to using rotors to perform rotation. I made screenshots for reference:
As the text suggest we can write $(e_1 + e_2)B_3$ to perform a rotation of vector $e_1 + e_2$ by 90 degrees counter-clockwise so in fact the result of $(e_1 + e_2)B_3$ should be $e_2 - e_1$ as shown in the figure, though I don't understand the derivation to get this result?
$ (e_1 + e_2)B_3 = e_1 B_3 + e_2 B_3 = e_1(e_1 e_2) + e_2(e_1 e_2)$
Where $ab$ in this context is the geometric product defined as:
$ab = a \cdot b + a \wedge b.$
I understand that because of associativity/communitivty you can write:
$ (e_1 + e_2)B_3 = e_1 B_3 + e_2 B_3 = e_1(e_1 e_2) + e_2(e_1 e_2) = (e_1 e_1) e_2 + e_1 (e_2 e_2)$
Is that correct? and how do I get to the final result $e_2 - e_1$?
The scalar product of a vector with itself is 1 (assuming the vector is of unit length) but I am not too sure what the vector product of a vector with itself is (it should be the vector 0,0,0 if the vector has three-dimensions, right?).
Could someone please help?
Orthogonal vectors anticommute. So $$e_2(e_1e_2) = -e_2(e_2e_1) = -(e_2e_2)e_1 = -(1)e_1 = -e_1$$