You haven't correctly answered my question yet: What vector stays the same when you rotate (by any $\theta\ne 0, 2\pi$, etc.) around the origin?
Thinking about this in proposition logic:
1. Sum of vectors is invariant under rotation
2. Sum of vector is also a vector
3. There is exactly one vector x that is invariant under rotation
4. (Missing)
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C. x is the sum of the vectors
