In mathematical terms what does parallel mean?

Answer 1

In two dimensions, to be parallel two lines must only have equal slopes.

y=x+2 and y=x-7 are parallel because they have the same slope (1).

In three dimenstions, this is also true, but the slopes must be equal in two directions. There are various ways of investigating this, but it is easiest to determine if two lines are parallel is if one can be expressed as a scalar multiple of the other. In three dimensions, lines can almost exclusively be drawn using vector-valued functions(especially for the scope of any class up to and including Calc 3). So, two three dimensional lines are parallel if one is a scalar multiple of the other, that is to say:

<x1(t), y1(t), z1(t)>=k<x2(t), y2(t), z2(t)> where "k" is some real number.

For example:

<t, t+1, 4-t> is parallel to <2t, 2t+2, 8-2t> because <2t, 2t+2, 8-2t>=2<t, t+1, 4-t>