Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$.
Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$.
Let $f_\xi: X\to BG$ be the classifying map of $\xi$.
Let $f_\eta:X\to G_n(\mathbb{R}^\infty)$ be the classifying map of $\eta$.
Let $i_*: BG\to G_n(\mathbb{R}^\infty)$ the induced map by inclusion $i:G\to GL(\mathbb{R}^n)$.
Are $i_*\circ f_\xi\simeq f_\eta$ homotopy equivalent?