In Hatcher's book on Vector Bundles, the tensor product of two vector bundles is defined through the gluing functions. But I need an example to understand it. So I think of the simplest case, the tensor product of two trivial line bundles.
Let $V_1$ and $V_2$ be two copies of $\mathbb R\times \mathbb R$, and $B = \mathbb R$. Let $\pi_i: V_i \rightarrow B, (x,y) \mapsto x$ for $i=1,2$. Are the gluing functions all identity maps? What is the tensor product of $(V_1,\pi_1)$ and $(V_2,\pi_2)$ as $B$-bundles and why? I guess that up to isomorphism, the result line bundle is still the trivial one, but I think what I need is something concrete.