With the outer product of the previous chapter we can span subspaces. It also enables us to compare lengths on a line, areas in the same plane, and volumes in the same space. We clearly have a need to compare lengths on different lines and areas in different planes. The nonmetrical outer product cannot do that, so in this chapter we extend our subspace algebra with a real-valued scalar product to serve this (geo) metric need. It generalizes the familiar dot product between vectors to act between blades of the same grade.

Then we carry the algebra further, and investigate how the scalar product and the outer product interact. This automatically leads to an inner product between subspaces of different dimensionality ...

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