Let $G\leq O(n)$ be a subgroup of orthogonal group. Let $\xi$ be a principal $G$-bundle. Let $\xi[\mathbb{R}^n]$ be the associated vector bundle.
If $\xi$ is not a trivial bundle, can we obtain that $\xi[\mathbb{R}^n]$ is not trivial?
If $\xi[\mathbb{R}^n]$ is a trivial bundle, can we obtain $\xi$ is a trivial bundle?