Dependece in the tensor product of two vector spaces

Question

Let $V$ be a $k$-vector space, and let $a, b, c, d, e, f \in V$. Suppose that $a \otimes b + c \otimes d = e \otimes f$. How do I show that either $a$ and $c$ are dependent or $b$ and $d$ are dependent? I know how to operate between tensor but so far I don't understant why it should be true as it is not specified anything about $a$, $b$, $c$, $d$, $e$, or $f$. I would appreciate any intuitive way of analyze this kind of problem.