Old English strica "line of motion, stroke of a pen" (related to strican "pass over lightly," see strike), from Proto-Germanic *strikon (cf. Middle Low German streke "stroke, line," Old High German, German strich, Gothic striks "stroke, line"), from PIE root *streig- (see strigil). Sense of "long, thin mark" is first found 1567. Meaning "a temporary run (of luck)" is from 1843.
1768, "to go quickly, to rush, run at full speed," respelling (by association with streak (n.)) of streek "to go quickly" (late 14c.), originally "to stretch oneself" (mid-13c.), a northern Middle English variant of stretch (v.). Streaking "running naked in public" first recorded 1973.