I've met some trouble in understanding Chern class. I first touch the Chern class in classyfying space of characteristic class. For $\pi:E\rightarrow X$ and we have such relationship:$$Vect^n_{\Bbb C}(X) \cong [X,Gr_n(C^{\infty})] \cong [X,BU(n)]$$ so for $f:M\rightarrow Gr_n(C^{\infty})\cong BU(n)$, we define the Chern class via pullbuck $f^*:H^*(BU(n),\Bbb Z)\rightarrow H^*(M,\Bbb Z)$. Intuitively speaking, this apprroach classfies the complex vector bundle.
It seems that Bott&Tu uses Grothendieck's approach by constructing line bundle and the first Chern class We then define the projectivization of vector bundle $\pi:P(E)\rightarrow M$, more precisely we have the taotological exact sequence $$0\rightarrow S \rightarrow \pi^{-1}E\rightarrow Q\rightarrow 0$$ here $\pi^{-1}E$ is the vector bundle over $P(E)$; $S$ is the universal bundle over $P(E)$ and $Q$ is the quotient bundle.
We then set $x=c_1(S^*)$ where $S^*$ is called hyperplane bundle which is dual to $S$ and Bott&Tu says the cohomology ring of $H^*(P(E))=H^*(M)[x]/(x^n+c_1(E)x^{n-1}+...+c_n(E))$
Here the Chern clss is defined via projective bundle and I can't undertstand it and its relationship between the way given by characteristic class?
More precisely, the relationship between Grassmannian manifold and projective bundle?