Vector weighting functions
Orthogonal but not orthonormal basis sets
Suppose we have two vectors from an orthonormal system, and . Taking the inner product of these vectors, we get
What if they aren't from a normalized system, so that
where the is the square of the length of and the symbol is one when k = n and zero otherwise? Well the general inner product of and becomes
You can interpret the as a weighting factor between the different directions so that different directions all end up in the units you would like. For example, suppose that the x and y directions were measured in meters, and the z direction was measured in centimeters, and you would like to use meters as your base unit. You could either convert the z dimensions to meters (probably simpler) or use a weighting function , and . In this sense, the system could be considered orthonormal with these units and this weighting arrangement.
Principle author of this page: Rob Frohne