If we now add a third orthonormal vector to our basis set, we generate the following geometrical objects:
From these objects we form a linear space of dimensions, defining multivectors as before, together with the operations of addition and multiplication. Most of the algebra is the same as in the 2-dimensional version because the subsets , and generate 2-dimensional subalgebras, so that the only new geometric products we have to consider are
These relations lead to new geometrical insights:
We should be quite clear, however, that we are using the symbol i to stand for a pseudoscalar, and thus cannot use the same symbol for the commutative scalar imaginary, as used for example in conventional quantum mechanics, or in electrical engineering. We shall use the symbol j for this uninterpreted imaginary, consistent with existing usage in engineering. The definition (2.17) will be consistent with our later extension to 4-dimensional spacetime.