This is just a matter of perseverance. You must use letters, not numbers, to prove this is true for all R3 vectors. For example, you could let u = (a,b,c), v = (d,e,f) and w = (g,h,i). Then, just set to work
computing the left hand side of the given inequality and the right hand side of the given inequality. Lots of expanding and collecting like terms.
For example, things like (a-g)^2 which must be expanded as (a-g)(a-g) = a^2 - 2ag + g^2.
You should discover that several of the terms are identical on both sides and can therefore be ignored. Focus on what is different.