Differential manifold connection with different notation

Question

We aim to prove that the set {V(n, 2) = {(a, b) \in R^n \times R^n : |a|^2 = |b|^2 = 1 \text{ and } <a,b> = 0} is a smooth (2n-3)-submanifold of R^n \times R^n.
This will be demonstrated by considering the function f :R^n \times R^n \rightarrow R^3 given by f(a, b) = (|a|^2, |b|^2, <a,b>) and showing that the derivative $Df(a, b)$ is a surjective linear transformation for all (a, b) $\in$ V(n, 2).
Should I use the implicit function theorem to prove the first one with derivative is the surjective linear mapping to R^3? and I dont know how to proof second part. Thanks for your help!