# Normals and the Inverse Transpose, Part 3: Grassmann On Duals

Welcome back! In the last couple of articles, we learned about different ways to understand normal vectors in 3D space—either as bivectors (part 1), or as dual vectors (part 2). Both can be valid interpretations, but they carry different units, and react differently to transformations.

In this third and final installment, we’re going leave behind the focus on normal vectors, and explore
a couple of other unitful vector quantities. We’ve seen how Grassmann bivectors and trivectors act as
oriented areas and volumes, respectively; and we saw how dual vectors act as oriented *line densities*, with
units of inverse length. Now, we’re going to put these two geometric concepts together, and find out
what they can accomplish with their combined powers. (Get it? Powers? Like powers of a scale factor?
Uh, you know what, never mind.)