I am quite confused about solving multi-variable equation systems with less than necessary lines. I understand I need to add more conditions but I just can't figure out how :
I have a set of three vectors (in $2$ dimensions), and by performing only rotations, I would like to obtain a given vector (target):
For example : $v_1 = (x_1, y_1), v_2 = (x_2, y_2), v_3 = (x_3, y_3)$ and $v_t = (x_t, y_t)$
let's call $rot_1, rot_2, rot_3$ three usual rotation matrices in $2D$ Euclidian space.
Then, my system is :
$\ rot_1 \times v_1 + rot_2 \times v_2 + rot_3 \times v_3 = v_t$
And I want to find the three corresponding angles: $\theta_1, \theta_2, \theta_3$
Of course, this system has infinitely many solutions, so I would like to restrict it by adding the following condition : $f : v_1, v_2, v_3 \to z \in \Re, f(rot_1 \times v_1, rot_2 \times v_2, rot_3 \times v_3) = min(f)$.
But I just don't know how to combine them together, am I doing it completely wrong ? I tried to take the problem from the math perspective instead of trying to brute force it by computer but for now it looks like a dead end.
Sorry, I realized my question is quite confusing, and really lacks details. I thank you all for your advices.