Basis of a subspace with a moving vector I have a problem and I don't know if there is a solution to it. 
I try to explain you an example which is simplified with respect to the scenario I have but it gives you a good idea of what my problem is.
You can look at the following figure to help your mind to think about the problem.



Imagine we have a set of 2 vectors $v_1$ and $v_2$ which span a plane $\pi$. The vectors $v_1$ and $v_2$ are fixed. Now I ask you:
I have a vector $v$ which is changing over time, which is already spanned by $v_1$ and $v_2$ and I want a new basis of the plane $\pi$ which includes the vector $v$ and one between $v_1$ and $v_2$. Because 2 vectors are enough to give us a basis of a plane, how to choose which vector between $v_1$ and $v_2$ to remove? 
Of course we can remove randomly one of them and take the other one. This will work, I mean I will have a basis of $\pi$ formed by $v$ and one between $v_1$ and $v_2$ but this won't work if, over time, $v$ becomes parallel to the one I chose. In this case I should have chosen the other one. 
My conclusion is that it is not possible to know this thing a priori and it is for this reason that the problem is not straightforward.
What do you think about it?
Thanks for the help.

I would suggest taking the vector $v$ and its orthogonal complement $v^\perp$ in the plane $\pi$, i.e., the vector 
$$v^\perp=av_1+bv_2$$ 
which is the solution to the equation 
$$v^\perp \cdot v = 0.$$ This will avoid the problem of a vector "vanishing" when $v$ is parallel to one of $v_1$ or $v_2$
