Technically the projective coordinate (3,2,1) should be exactly the same as (6,4,2), and every nonzero multiple thereof. So it’s not really correct to say that

Technically the projective coordinate (3,2,1) should be exactly the same as (6,4,2), and every nonzero multiple thereof. So it’s not really correct to say that (x, y, 0) represents a vector, or that adding these projective coordinates is vector addition. Vector addition is represented by an identity matrix with the x, y coordinates in the rightmost column.