Vector manipulation in spherical coordinates

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I have a geology background therefore apologies if I don't use the correct terminology to describe my problem.

I have a line with an azimuth of 199 degrees and an inclination (from horizontal) of 60 degrees. The orientation of this line was determined from an image that also has an orientation: azimuth 220 degrees and inclination of 88 degrees (from horizontal).

I want to correct the image so that it is vertical (by rotating it along an axis that is at right angles to its azimuth). My question is, how can I calculate what the modified orientation of this line will be?

Notes:

  • The line describes the maximum slope on an inclined 2D plane.
  • The magnitude of the vector is not of interest therefore I've been treating it as 1.
  • To better conceptualise the problem, the line I am describing is the line in red on the below image, its orientation is described by a dip angle and dip direction (in geological terminology).

enter image description here