I've drawn a representation of the vector i described.
It's magnitude is simply it's length. And as the vector form's a right-angled triangle it's magnitude is simply as follows:
squareroot(square(x) + square(y))
Yes, a vector would have fields X and Y.
Not sure what you mean about making it an arror pointing to it.
The arrow is to help with understanding only (as you said you think of it as a line).
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