Oriented volume in 3-space requires three vectors that are linearly independent. If these are the vectors A, B, and C then linear independence means that any of these given vectors is not a scalar multiple of the others, that is A=��B or B=��C or A=��C where �� is a scalar of real numbers, which can be rational or irrational and �� is never identically zero. Since any vector is defined as having a magnitude and a direction, if A=��B then A and B are both pointing along the same direction. Therefore, linear independence implies that all three vectors A, B, and C are pointing in different directions. For optimized spanning of 3-space three orthogonal unit vectors must be defined. For example, the three unit vectors found in a Cartesian coordinate system. If any two unit vectors are not orthogonal then by the law of vector’s components projection, any vector can be projected into the others making them linearly dependent by defining subunit vectors. The subunit vector is properly a projection of the lesser magnitude vector to the direction of the greater magnitude vector.

Although three orthogonal unit vectors are sufficient to span 3-space by their directional projections, at most it spans either a right-handed system or a left-handed system. To completely span the totality of the spacetime continuum requires both right-handed and left-handed systems. Their combined spacetime topologies give eight directional invariance properties. These are equivalent to the reality of gluons in the physical theory of quantum chromodynamics of three color forces as three unit vectors spanning the infinitesimal region required by a quantum theory of the spacetime continuum. This theory asserts the existence of spacetime charges: H-plus and H-minus, or as A and B vectors. Then the C vector can always be expressed as a linear combination of A and B and the oriented volume is always zero satisfying the axiom of linear dependence.