In three-dimensional space, lines can be classified into four main categories based on their relative positions:
Consider the lines: $L_1: (x, y, z) = (1, 2, 3) + t(1, 1, 1)$ $L_2: (x, y, z) = (1, 2, 3) + s(2, 2, 2)$
These lines are coincident because $L_2$ can be obtained from $L_1$ by substituting $t = 2s$.
The lines: $L_1: (x, y, z) = (1, 2, 3) + t(1, 1, 1)$ $L_2: (x, y, z) = (4, 5, 6) + s(1, 1, 1)$
are parallel because they have the same direction vector $(1, 1, 1)$ but different points of origin.
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