In page-149 of Tristan Needham's Visual Differential Geometry, the following picture is given:
I was thinking of this interpretation, and I think there is a problem with it. Suppose we have it that the length of vector is less than the spacing of vector? Then it would be that the one form should give zero when evaluated on a vector as it cuts through no planes, but if you actually evaluate the formula you get something non zero.
Eg: Consider the cartesian basis form $dz$, if we have a vector $v =.1\hat{k}$, then by the above interpretation, clearly $dz(v) $ should be zero since $v$ is too small to cut through any planes. But in actuality when we evaluated $dz(v)$ , we get $.1$
So am I misinterpreting the interpretation or are these type of issues known to exist with it?