= 0
The general six-vector f is composed from the vectors φ, φ* in the following way:—
f = ρφ + ρ* φ*,
ρ and ρ* denoting the contents of the pieces of mutually perpendicular planes composing f. The “conjugate Vector” f* (or it may be called the complement of f) is obtained by interchanging ρ and ρ*.
We have
f* = ρ*φ + ρφ*
We can verify that
fy z* = fx l etc.
and f² = ρ² + ρ*², (ff*) = 2ρρ*.
| f |² and (ff*) may be said to be invariants of the six vectors, for their values are independent of the choice of the system of co-ordinates.