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 length of x is 6.
For multiple vectors, the total distance is the sum of each vectors' distance. Continuing with the example for x, if y were [2 -1 0], then their total distance would be 14 + 5 = 19.
The total distance can also be thought of as the distance of the singular vector z resulting from x+y. If x were [1 2 3] and y were [2 -1 0], then z would be [3 1 3] and the distance of z is 19.
The Pythagorean theorem continues to hold in higher dimensions. Importantly though, the squared distance of a singular vector x is the sum of each components' square. For either of the above xs, the squared distance is 14. This can be notated as xTx. For two vectors x and y, the Pythagorean theorem is written as xTx + yTy = (x+y)T(x+y).
Because it is simple to sum each vectors' distance, the Pythagorean theorem isn't used to compute total distance in higher dimensions. Rather, it leads to the proof for orthogonality.
Distance of a vector is commonly notated as ||z||.