From mathematics we know that rotations of of a vector in three-dimensional Euclidean space are described by representations of SO(3) group, which can be parametrized using three angles (one for each axis). However, it seems to me that every possible vector can be rotated to any other vector (of the same length) by using just rotation about two of the angles.
Here's an example: start with vector of unit length lying along x-axis. Now every possible vector of unit length can be deformed from this vector by rotating first about z-axis and then about x-axis -- or at least it seems to me so after drawing pictures and thinking about it.
So can you give me a counter-example of two vectors of the same length that can't be deformed to each other by using just two rotations? Or if such example doesn't exist, why do you need three parameters to describe rotations?
From group theoretical perspective I understand why SO(3) group needs three parameters, but thinking in terms of rotations of a vector I can't figure it out.
You are correct that a vector can be taken to another using two angles. Vectors don't have any structure along their axis. The third angle is rotation around the vector, which is needed if there is a yaw orientation that matters.