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1. The only definition I have seen depends on the basis you use for your coordinate system, and it defines positive vectors as those with all positive coordinates, and negative vectors having all negative coordinates. It is useful in practice to define non-negative vectors, those whose coordinates are all non-negative.
2. Equal vectors are defined as two vectors having same magnitude and direction. They may or may not have same initial point. So if the initial point is same then they will also equal vector if both vectors satisfying condition of equal vector.
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