So yes, this is homework, but for the class that I'm TA'ing. The question is something like:
Given two lines in 2D homogeneous representation ($\mathbb{R}^{2+1}$): $\bf a$ and $\bf b$, their intersection is $\bf p = a \times b$ (so far so good)
"The sine of the angle of their intersection is the homogeneous weight of this point." Prove this.
I assumed that by homogeneous weight, what is meant is the third value ($\lambda$) of the point $\bf p$, where $\mathbf{p} = (\lambda p_1, \lambda p_2, \lambda) $, then what needs to be proven is that $p_3 = a_1b_2 - a_2b_1 = \sin \theta$ with $\theta$ the angle between those lines. But I can't figure this out.
Aside: This would have been an easier question if in stead of homogeneous weight they asked for the magnitude of the intersection, then it is solved by assuming that $\bf a$ and $\bf b$ are normalized, and using $\mathbf{a} \times \mathbf{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin \theta \ \mathbf{n}$, but I'm not sure if that's what is meant.