How Am I Wrong? Geometric Algebra Question I'm currently teaching myself Geometric Algebra out of the textbook Geometric Algebra for Physicists, and there's an example I get wrong. In the example, they show that $(\mathbf{a}\wedge\mathbf{b})^2=-\mathbf{a}^{2}\mathbf{b}^2sin^2{θ}$

My Work:
$(a \wedge b)^2 = (a\wedge b)(a \wedge b) = (a\wedge b)(-b \wedge a) = (ab - a \cdot b)(-ba + a\cdot b)= $
$= a^2 b^2 - (a\cdot b)^2 + a\cdot b(ab - ab)$
$= a^2 b^2 - (a\cdot b)^2$
$= a^2 b^2 - (|a||b| \cos(\theta))^2$
$=  a^2 b^2 -  a^2 b^2 \cos^2(\theta)$
$= a^2b^2(1 - \cos^2(\theta))$
$= a^2b^2 \sin^2(\theta)$

I almost get the right answer, but keep getting the negative of the book's answer. What am I doing wrong?

From the first to second row:
$(ab- a\cdot b)(-ba + a \cdot b)  \\=-abba+(a \cdot b)ab+(a \cdot b)ba-(a \cdot b)^2 \\=-a^2 b^2 +(a \cdot b)(ab+ba) -(a \cdot b)^2
\\= -a^2 b^2+2(a \cdot b)^2-(a \cdot b)^2
\\=-a^2 b^2 +(a \cdot b)^2$
which gives you the overall minus sign that you were after.
