SOLUTION: How do I prove the theorem "Through a given point of a given line there is only one plane perpendicular to the line"?

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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.