Question

How to prove `ab+cd<=sqrt((a^2+c^2)(b^2+d^2))` given a, b, c, d, are any four positive numbers?

Answer 1

See below.

Explanation:

Given two vectors

`vec u = (a,c)`
`vec v = (b,d)`

we know that

`<< vec u, vec v >> = norm(vec u) norm(vec v) cos phi rArr << vec u, vec v >> le norm(vec u) norm(vec v)`

or

`ab+cd le sqrt(a^2+c^2)sqrt(b^2+d^2) le sqrt((a^2+c^2)(b^2+d^2))`