_{12}and e

_{23}are weakly perpendicular, e

_{12}and e

_{34}are strongly perpendicular. Once this distinction is made then angles may be calculated.

Given normalized subspaces A and B with grades a and b the weak angle is cos theta = ||AB<|a-b|>|| while the strong angle is sin tau = ||AB<a+b>||. In 3D they are usually the same, the exception being that the strong angle between two planes or volumes is always zero.

These notions may be extended to multivectors. Angles seem to me to have little meaning here so instead one may make use of the identity that if M and N are normalized multivectors with maximum grades m and n then MN<0>

^{2}+ MN<1>

^{2}+ MN<2>

^{2}... MN<m+n>

^{2}= 1. The more weight have the lower order terms, the more the two multivectors have in common.

In 3D to find the subspace between two subspaces take the average. This doesn't necessarily work in higher dimensions. Suppose our two subspaces are e

_{12}and e

_{34}. The normalized average of these will have strong and weak angles of pi/4 w.r.t. both e

_{12}and e

_{34}but is not a subspace. The subspaces that come closest to satisfying our requirement are e

_{13}and e

_{24}. Either is weakly perpendicular to both e

_{12}and e

_{34}.