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How to make a line intersect/pierce a surface, please help


I have set up a plane and marked a construction line as shown. I want to make that line intersect with the surface, so I can sketch from that point of intersection. Can anyone please help me? I've tried adding relations etc. but I cant get it to work.
EDIT: Just tried the Split line tool, and it does the job but leaves a line across the surface, is there a way to get rid of this line?
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