Question

Is (ac+bd)^2 <= (a^2+b^2)(c^2+d^2) for any real numbers a,b,c,d ?

thanks

Answer 1

if you unfold both sides:
\[a ^{2}c ^{2}+2acbd+b ^{2}d ^{2}=a ^{2}c ^{2}+a ^{2}d ^{2}+b ^{2}c ^{2}+b ^{2}d ^{2}\]
simplify:

Answer 2

\[2acbd=a ^{2}d ^{2}+b ^{2}c ^{2}\]
which could be presented as:
\[(ad+bc)^{2}=0\]
which gives you:
ad=-bc
this is condition when your original equation is true