Distance
Euclidean distance in higher dimensional space is useful for normalization.
Contents
2 Dimensions
In 2 dimensions, the distance between two Cartesian points is calculated by the Pythagorean theorem: x2 + y2 = z2. This is sometimes referred to as the Pythagorean distance.
When this concept is expanded to complex numbers (but still in 2 dimensions), a common notation is |x - y| to emphasize that the distance must be normalized to an absolute value.
n Dimensions
For a singular vector x, the distance is the sum of each components' absolute value. If x is [1 2 3] or [-1 -2 -3], the distance of x is 6. Note that distance of a vector x is notated as ||x||.
The Pythagorean theorem continues to hold in higher dimensions. Note that the theorem calls for squared distances: x2. In other words, the squared distance of x is the sum of each component squared. For either of those x vectors, the squared distance is 14. Recall though that in matrix notation, multiplying two vectors creates a matrix; true vector multiplication is notated as xTx.
Plugging this into the Pythagorean theorem then, if y were [2 -1 0] (with squared distance of 5), then the squared distance of z is 19. The actual distance is √19.
This can be double checked using the given values of x and y, because vectors can be added directly. x + y = z, so [1 2 3] + [2 -1 0] = [3 1 3]. And with that calculated z, it's clear again that the squared distance of z is 19.
For two vectors x and y, the Pythagorean theorem can be written as xTx + yTy = (x+y)T(x+y). This formulation leads to the test for orthogonality.