If $p_0, p_1, p_2$ are three distinct points in space, then what does the cross product $$n = (p_0 - p_1) \times (p_0 - p_2)$$ mean geometrically?
I'm having a little trouble visualizing this in $3$-space.
Taking the cross product obviously gets me the orthogonal vector to $(p_0 - p_1)$ and $(p_0 - p_2)$, but I'm still not sure how to visualize this... any suggestions?
I'd rewrite $n$ as $$n = (p_1 - p_0) \times (p_2 - p_0),$$ so that we can visualize $p_1 - p_0$ and $p_2 - p_0$ as vectors based at $p_0$. Then, $n$ is mutually orthogonal to both of these vectors and points in the direction specified by the right-hand rule. The length of $n$ is $|p_1 - p_0| |p_2 - p_0| \sin \theta$, where $\theta$ is the angle between the two vectors.