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The zero vector is both parallel and perpendicular to any other vector. V.0 = 0 means zero vector is perpendicular to V and Vx0 = 0 means zero vector is parallel to V.
The zero vector is not perpendicular to all vectors, but it is orthogonal to all vectors.
Their directions are perpendicular.
I think you meant to ask for finding a perpendicular vector, rather than parallel. If that is the case, the cross product of two non-parallel vectors will produce a vector which is perpendicular to both of them, unless they are parallel, which the cross product = 0. (a zero vector)
If all the components of a vector are zero, the magnitude of the vector will always be zero.