Parallel lines in 3D: distinguishing them from coincident lines
Test parallel directions and point membership to distinguish distinct parallel lines, coincident lines, and skew lines.
Updated: October 22, 2026
Lines with parallel direction vectors need not be distinct parallel lines: they may coincide. In Oxyz, first compare their directions, then test a point to distinguish these cases.
Learning goals: recognize distinct parallel lines; use vector tests without dividing components; distinguish parallel, coincident, and skew lines.
Prerequisites: nonzero direction vectors, the cross product, and parametric equations.
On this page: Definitions · Method · Examples · Practice · Further study
1. Parallel directions alone do not establish distinct parallel lines
Let $d_1$ pass through $M_1$ with direction vector $\vec u_1\ne\vec0$, and $d_2$ through $M_2$ with direction vector $\vec u_2\ne\vec0$. In this lesson, $d_1\parallel d_2$ means distinct lines with parallel directions and no common point. Coincident lines are treated separately.
If $\vec u_1,\vec u_2$ are parallel, the lines are either distinct and parallel or coincident. Test a point of one line on the other: if $M_2\in d_1$, they coincide; if $M_2\notin d_1$, they are distinct and parallel.
2. Vector tests for direction and position
Set $\vec w=\overrightarrow{M_1M_2}$. The following criterion avoids coordinate fractions:
The first condition says that the direction vectors are parallel. The second says that $M_2$ does not lie on $d_1$: points of $d_1$ have the form $M_1+t\vec u_1$. If the second cross product is also $\vec0$, the lines coincide.
To see why testing one point suffices, suppose the lines have parallel directions and a common point $I$. Both are the set of points $I+t\vec u_1$, with $t\in\mathbb R$, so they coincide. Thus distinct lines with parallel directions cannot intersect.
State a point and direction vector for each line, check that each direction differs from $\vec0$, then examine the two cross products. This method still works when one or two direction-vector components are $0$; ratios with denominator $0$ are unnecessary.
3. Worked examples
Example 1. Checking direction and point membership
Consider the lines
The normal vectors of the planes defining $d_1$ are $(1;1;1)$ and $(1;-3;0)$. Their cross product is $(3;1;-4)\ne\vec0$, so take $\vec u_1=(3;1;-4)$. For $d_2$, choose $M_2(1;-1;1)$ and $\vec u_2=(-3;-1;4)=-\vec u_1$.
Substituting the parametric equations of $d_2$ into the first plane equation gives
Thus no point of $d_2$ lies in the first plane, so $d_1$ and $d_2$ do not intersect. Their direction vectors are parallel, hence $d_1\parallel d_2$. There is no need to reverse the parameter sign when substituting.
Example 2. Parallel directions but coincident lines
Let $d_1$ pass through $O(0;0;0)$ with direction vector $(1;2;-1)$, and $d_2$ through $B(2;4;-2)$ with direction vector $(-2;-4;2)$. The direction vectors are parallel and $\overrightarrow{OB}=2(1;2;-1)$, so $B\in d_1$. Thus the lines coincide; they are not distinct parallel lines.
Example 3. Nonintersecting lines may be skew
The lines $d_1:(x;y;z)=(t;0;0)$ and $d_2:(x;y;z)=(0;s;1)$, with $t,s\in\mathbb R$, do not intersect because their $z$ coordinates are $0$ and $1$, respectively. However, their directions $(1;0;0)$ and $(0;1;0)$ are not parallel. The lines are skew. Nonintersection alone does not imply parallelism.
4. Practice
- Classify the pair $d_1:(x;y;z)=(t;0;0)$ and $d_2:(x;y;z)=(s;2;0)$.
- Classify the pair $d_1:(x;y;z)=(t;1+t;2)$ and $d_2:(x;y;z)=(2+3s;3+3s;2)$.
- Classify the pair $d_1:(x;y;z)=(t;0;0)$ and $d_2:(x;y;z)=(0;1;s)$. All parameters range over $\mathbb R$.
Hints and answers
- The lines are distinct and parallel: they share direction vector $(1;0;0)$, but $(0;2;0)$ does not lie on $d_1$.
- The lines coincide: the second direction vector is three times the first, and $(2;3;2)$ lies on $d_1$ when $t=2$.
- The lines are skew: their different $y$ coordinates rule out intersection, and their directions $(1;0;0)$ and $(0;0;1)$ are not parallel.
Common mistakes: concluding parallelism solely from parallel direction vectors; calling nonintersecting lines parallel without comparing directions; dividing a coordinate by $0$; using one common parameter value to test intersection of independently parametrized lines.
5. Further study and resources
Direction vectors of lines (English lesson) · Equations of lines in space (English lesson) · Intersecting lines (English lesson) · Skew lines (English lesson).
Original resource on Scribd (external link).
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