A Householder reflection of a vector $v$ along a direction $n$ is given by the formula \begin{eqnarray} v' = v - 2 \frac{n \cdot v}{|n|^2} n. \end{eqnarray}
If $n$ is unitary then $v'=v-2 (n \cdot v) n$. Clearly for $v=(2,3)$ and $n=(0,1)$ we find that $v'=(2,-3)$ which in fact is the Householder reflection of $(2,3)$ along the vector $(0,1)$.
According to Geometric Algebra (GA) a Housholder reflection is defined as $-nvn^{-1}$. Since $n$ is unitary $n^{-1}=n$ and we can write $v'=-nvn$. However when I compute this formula with the vectors above, I get different results.
Let me explain: \begin{eqnarray} n v = (0,1) \cdot (2,3) + (0,1) \wedge (2,3) = 3 - 2 e_1 \wedge e_2. \end{eqnarray} and so
\begin{eqnarray} v' = -nvn^{-1} = -(3 -2 e_1 \wedge e_2) n = -3n + 2 e_1 \wedge e_2 n. \end{eqnarray}
My concern is that, since $e_1 \wedge e_2$ is equivalent to a 90 degree counterclockwise rotation then we find that the second (last) term here is $(-2,0)$ and so the result is $v'=-3(0,1)+2(-1,0)=(-2,-3)$ and not $(2,-3)$ as expected.
Probably I have a stupid error such a sign somewhere or a deep problem on understanding the geometric algebraic product.
I would appreciate any help on this matter.
Thanks.
The problem is apparently with your interpretation, because the math is working out. $$ -nvn^{-1} = -(3 -2 e_1 \wedge e_2) n = -3n + 2 e_1 \wedge e_2 n=-3e_2+2e_1e_2e_2=2e_1-3e_2$$.
I’m not even sure why you were using the product identity with the wedge. You may as well just compute it directly:
$$ -nvn =-e_2(2e_1+3e_2)e_2=-2e_2e_1e_2-3e_2e_2e_2=2e_1-3e_2 $$
It seems to me that multiplication by $e_1\wedge e_2$ on the left creates a clockwise rotation by 90, since it maps $e_2$ to $ e_1$ and $e_1$ to $-e_2$.