In an inner product space V then |〈v,w〉| ≤ ∥v∥×∥w∥ for any v,w ∈V, with equality if and only if v,w are linearly dependent. Thus, the angle between non-zero vectors v,w can be defined as arccos (〈v,w〉/(∥v∥×∥w∥)).
Statistics
If X and Y are two random variables thenwith equality if and only if P(aX = bY) = 1 for some real constants a and b, at least one of which is non-zero.