Considering the Axis of a Straight Wire: Are There Other Considerations?

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When one comes across the sentence 'axis of a straight wire', how can one consider the orientation of the axis?

Commonly, the axis of a straight wire can be considered to be a line parallel(along the wire) to the wire itself.

Can 'axis of a straight wire' also mean a line that is perpendicular to the wire? I doubt it. But, I wish to know if the latter is also true.
 
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I would interpret it as a longitudinal axis along (parallel to) the wire passing perpendicularly through the center of its (circular) cross section, sometimes called the 'axial' longitudinal axis. (As you also have apparently interpreted it).
I would guess that if one were looking at an axis perpendicular to the wire, the sentence would state such, for example, a yoyo and cord rotating about an axis peppendicular to the plane of rotation.
 
You can have an axis perpendicular to the straight wire, but a point external to the wire must be specified in order to determine which set of axes you are concerned with.