# A possible kind of \$K\$ theory via comparison of sphere bundles associated to given vector bundles

Let $$E$$ be a vector bundle on a topological space $$X$$.Thanks to Allen Hatcher’s book “Vector Bundles and K theory”, the construction of sphere bundle $$S(E)$$ can be done without any inner product on fibers. It is a result of the equivalent relation on each punctured fiber: $$vsim w$$ if $$v=lambda w$$ for some positive scalar $$lambda$$.

We define an equivalent relation on the set of all vector bundles over $$X$$ as follows: $$E$$ is equivalent to $$F$$ if their corresponfing sphere bundles $$S(E)$$ and $$S(F)$$ are isomorphic fiber bundles.

Is the above equivalent relation compatible to direct sum and tensor product of vector bundles in the following sense;

If $$E_1,E_2$$ are equivalent bundles and $$F_1,F_2$$ are also equivalent bundles in the above sense, are $$E_1oplus F_1, E_2 oplus F_2$$ are equivalent bundles? What about the tensor product, instead of direct sum?
Can this equivalent relation define a kind of $$K$$ theory?