If the vectors ⃗u =(1 2) and ⃗v =(2 −4) are orthogonal with respect to the weighted inner product ⟨⃗u, ⃗v⟩ = w1u1v1 + w2u2v2, what must be true about the weights w1, w2?

Answer 1

By definition, two vectors u and v are orthogonal with respect to an inner product < > if <u,v>=0. In our case we must have 2w_1-8w_2=0 or equivalently, w_1=4w_2.