Assume that $P\to M$ is a principal $G$-bundle where $G$ is some (compact) Matrix group. Let $\rho\colon G \to \operatorname{Gl}(\mathbb{R}^n)$ be the tautological representation and $\rho^\prime\colon G\to \operatorname{Gl}(V)$ some other representation.

Let $$ E = P \times_{\rho}\mathbb{R}^n, \qquad \text{and} \qquad F = P\times_{\rho^{\prime}}V $$ be the associated vector bundles.

Using the Chern-Weyl theory and the usual definitions can associate characteristic classes to $P$, $E$ and $F$.

Are they equal?

To make this less vague, assume $G$ is a complex Matrix group and let`s look at $c_1(P)$, $c_1(E)$ and $c_1(F)$. I assume that $c_1(P) = c_1(E)$. Is this also true for $c_1(F)$?

To make it even less vague, consider the Hopf bundle $\mathbb{S}^3 \to \mathbb{S}^2$, and let $\rho_n\colon U(1)\to \operatorname{Gl}(\mathbb{C})$ be the usual irreducible representations for $n\in\mathbb{Z}$. Then $c_1$ of these vector bundles are a complete invariant. Assuming $c_1(\operatorname{Hopf}) = -1$, what is $$c_1\left(\operatorname{Hopf}\times_{\rho_n}\mathbb{C}\right)? $$

I apologize if this is too trivial.