A negative of a vector represents the direction opposite to the reference direction. It means that the magnitude of two vectors are same but they are opposite in direction.

For example, if A and B are two vectors that have equal magnitude but opposite in direction, then vector A is negative of vector B.

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Yes, a vector which has zero magnitude is also a vector in case of two vectors travelling in opposite directions with equal magnitudes. At this case, the resultant vector has zero magnitude but it is still a vector. We call it a null vector.