In $3$-dimensional space, if lines $L_1$ and $L_2$ do not intersect, then they must be parallel. True or false?
The proposition is false.
Two lines are parallel if their direction vectors are dependent (one of them is a multiple of the other). If $u$ and $v$ are two non-zero vectors in 3-dimensional space, there exist lines with direction vectors $u$ and $v$ that don't intersect. If $L$ is the line passing through $(0,0,0)$ and parallel to $u$ and $L_1, L_2, L_3$ are the lines parallel to $v$ passing respectively through $(1, 0, 0), (0,1,0), (0,0,1)$, then one of the $L_i$s does not intersect $L$.