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## The Algebra of 3-Space

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

and

These relations lead to new geometrical insights:

• A simple bivector rotates vectors in its own plane by , but forms trivectors (volumes) with vectors perpendicular to it.
• The trivector commutes with all vectors, and hence with all multivectors.
The trivector also has the algebraic property of being a square root of minus one. In fact, of the eight geometrical objects, four have negative square , , , . Of these, the trivector is distinguished by its commutation properties, and by the fact that it is the highest-grade element in the space. Highest-grade objects are generically called pseudoscalars, and is thus the unit pseudoscalar for 3-dimensional space. In view of its properties we give it the special symbol i:

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.

Mark Ashdown