In the 3-dimensional space [X.sup.(3)], for any pair of vectors, the Cauchy-Schwarz inequality  holds, which for the vectors (5) gives the implication:
where [phi](t) is instantaneous angle between vectors of the IV of current and voltage in the arithmetic 3-dimensional space [X.sup.(3)] at time t.
In this article, we investigate the radial part of Laplace's equations in dimensions both lower and higher than three and interpret what their solutions represent in 3-dimensional space. Consider an n-dimensional Euclidean space given by the linear coordinates [x.sub.1], [x.sub.2], [x.sub.3], ..., [x.sub.n] [-[infinity] < [x.sub.1], [x.sub.2], x, ..., [x.sub.n] < [infinity]].
We can now examine the physical examples which the solutions to Laplace's equations represent in our familiar 3-dimensional space. First, for n = 1, Eqs.
Clearly, this case belongs to an infinite line charge distribution in 3-dimensional space, where the electric field diminishes inversely as the radial distance from the line charge.
These are identified as the potential and electric field, respectively, of the electric dipole in 3-dimensional space.
VR in this sense is a medium where a user can operate within a realistic representation of 3-dimensional space, in real time.
The user has an additional navigation "remote control," with arrows to steer and change the orientation in 3-dimensional space.