For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy
$$
f(\xi): B\longrightarrow BG, 
$$
$f(\xi)\in [B;BG]$, and the composition up to homotopy
$$
g(\xi): B\overset{f}{\longrightarrow} BG\overset{i}{\longrightarrow}BO(n)\overset{j}{\longrightarrow}BO$$
where $BO=\lim \_{n\to\infty} BO(n)$ and $g(\xi)\in [B;BO]$.


Suppose we have two such $n$-dimensional vector bundles $\xi\_1$, $\xi\_2$.


**Question:** Are there any formulas
$$
g(\xi\_1\oplus\xi\_2)=? \text{ in terms of } g(\xi\_1), g(\xi\_2)?
$$
And
$$
f(\xi\_1\oplus\xi\_2)\in [B,BG\times BG]=?\text{ in terms of } f(\xi\_1), f(\xi\_2)?
$$



classifying maps of Whitney sums of vector bundles
$BO(n)$ is the infinite-dimensional Grassmannian $Gr(n,\infty)$ of $n$-planes in ${\mathbf R}^\infty$. There is a natural direct sum operation
$$\oplus\colon Gr(n,\infty)\times Gr(m,\infty)\to Gr(n+m,\infty)$$
(just taking the direct sum of linear subspaces) and it gives you the desired map 
$$BO(n)\times BO(m)\to BO(n+m).$$


