In two dimensions, two lines have only three possible relationships: they are equal (the same line), parallel but distinct, or they intersect at exactly one point. Three dimensions introduces a fourth possibility that doesn’t exist in 2D geometry.
When working with two lines in three-dimensional space, exactly one of these four relationships must be true:
Equal lines: The lines are identical (same direction, same points)
Parallel lines: The lines have the same direction but never meet
Intersecting lines: The lines cross at exactly one point
Skew lines: The lines are not parallel and do not intersect
skew lines
Two lines are skew if they are not parallel but do not intersect. This relationship is unique to three-dimensional space and cannot occur in a plane.
Figure 5. In three dimensions, it is possible that two lines do not cross, even when they have different directions.
To classify the relationship between two lines, we need to answer two key questions:
Are the direction vectors parallel? (Do the lines have the same or opposite directions?)
Do the lines share any points? (Is there a common point on both lines?)
The answers to these questions determine the relationship.
Figure 6. Determine the relationship between two lines based on whether their direction vectors are parallel and whether they share a point.
line classification
Direction vectors parallel AND lines share a point → Lines are equal
Direction vectors parallel AND lines share no points → Lines are parallel but distinct
Direction vectors not parallel AND lines share a point → Lines intersect
Direction vectors not parallel AND lines share no points → Lines are skew
In 2D, if two lines aren’t parallel, they must intersect. In 3D, they might “miss” each other entirely—imagine one line running east-west on the ground floor of a building while another runs north-south on the second floor.
For each pair of lines, determine whether the lines are equal, parallel but not equal, skew, or intersecting.
Line [latex]L_1[/latex] has direction vector [latex]{\bf{v_1}}=\langle2,1,1\rangle[/latex]; line [latex]L_2[/latex] has direction vector [latex]{\bf{v_2}}=\langle1,3,-2\rangle[/latex]. Because the direction vectors are not parallel vectors, the lines are either intersecting or skew. To determine whether the lines intersect, we see if there is a point, [latex](x, y, z)[/latex], that lies on both lines. To find this point, we use the parametric equations to create a system of equalities:
There is no single point that satisfies the parametric equations for [latex]L_1[/latex] and [latex]L_2[/latex] simultaneously. These lines do not intersect, so they are skew (see the following figure).
Figure 7. Lines [latex]L_1[/latex] and [latex]L_2[/latex] are skew.
Line [latex]L_1[/latex] has direction vector [latex]{\bf{v_1}}=\langle1,-1,1\rangle[/latex] and passes through the origin, [latex](0, 0, 0)[/latex]. Line [latex]L_2[/latex] has a different direction vector, [latex]{\bf{v_2}}=\langle2,1,1\rangle[/latex], so these lines are not parallel or equal. Let [latex]r[/latex] represent the parameter for line [latex]L_1[/latex] and let [latex]s[/latex] represent the parameter for [latex]L_2[/latex]:
Solve the system of equations to find [latex]r=1[/latex] and [latex]s=-1[/latex]. If we need to find the point of intersection, we can substitute these parameters into the original equations to get [latex](1, -1, 1)[/latex] (see the following figure).
Figure 8. Lines [latex]L_1[/latex] and [latex]L_2[/latex] intersect at [latex](1, -1, 1)[/latex].
Lines [latex]L_1[/latex] and [latex]L_2[/latex] have equivalent direction vectors: [latex]{\bf{v}}=\langle6,-2,3\rangle[/latex]. These two lines are parallel (see the following figure). Figure 9. Lines [latex]L_1[/latex] and [latex]L_2[/latex] are parallel.
