Given the 4 points describing 2 lines, how do you quickly calculate if line A is facing towards or away from line B?

Similarly, how to check if line A is towards or away from quadratic bezier curve B?

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Take the dot product of the unit vector of the candidate line and the unit normal pointing in the direction you wish for the other (there are two unit normal vectors).
If the dot product is positive, it’s “away”; “towards” is negative.
If you don’t know vectors, get an intro text and read up on them.