Isomorphic subbundles of codimension $1$ imply isomorphic quotients?

Question

This is a follow-up of this question.

Let $\tilde V, \tilde W$ be two isomorphic real vector bundles of rank $k$ over a smooth manifold $M$.

Suppose there exist isomorphic subbundles $V \subseteq \tilde V,W \subseteq \tilde W$ of rank $k-1$, i.e $V \cong W$.

Question: Is it true that $\tilde V/V \cong \tilde W/W$? (are the quotients isomoprhic?)

If $\tilde V,\tilde W$ are orientable, then the answer is positive, as explained here. I wonder what happens if we don't assume they are orientable.

Answer 1

Yes, if you assume $\tilde{V}$ and $\tilde{W}$ are isomorphic then the quotients are isomorphic. This follows from a computation with Stiefel-Whitney classes: we have

$$w(\tilde{V}) = w(V) w(L_1) = w(\tilde{W}) = w(W) w(L_2)$$

where $L_1, L_2$ are the quotients, from which it follows that $w(L_1) = w(L_2)$, hence $L_1$ and $L_2$ are isomorphic. With no hypotheses on the codimension what we get is that the quotients have the same Stiefel-Whitney classes, which is weaker in general than being isomorphic, but is equivalent for line bundles.