Question 388289
You can prove the converse, that the vector normal (you use perpendicular, not exactly the same) to a plane is unique in direction.  Then any line parallel to the vector thru a chosen point is normal to the plane, and not normal to any plane not parallel to the given plane.
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As an alternative, choose a point on a line in 3-space.  Find 2 lines not 180º apart perpendicular to the 1st line.  Those 2 lines define a plane.