TITLE: How do we define the distance between two lines or planes that intersect?
QUESTION [7 upvotes]: How do we define the distance between two lines or planes that intersect? Since it cannot be defined injectively for every point, is it defined at all in such a case?

REPLY [5 votes]: In this situation it is better to use the distance on the Grassmann manifold. First, given a set $E \subset \mathbb R^p$ and $v \in \mathbb R^p$, let
$$
d(v,E)=\inf \big\{\lVert v-w \rVert: w \in E \big\}.
$$
Then, given subspaces $E,F \subset \mathbb R^p$, we define
$$
d(E,F)=\max \bigg\{ \max_{v \in E, \lVert v \rVert=1} d(v,F), \max_{w \in F, \lVert w \rVert=1} d(w,E) \bigg\}.
$$
PS: We cannot use angles without anything else simply because they do not define a distance!