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In two dimensions, two lines can either be parallel, the same line, or intersect exactly once. In three dimensions, lines have similar behavior, but with an added level of complexity due to the additional dimension.
Discussion
Recall that in two dimensions, a pair of lines is parallel if the lines have the same gradient but no points in common, like two train tracks running next to each other.
How do you think this translates to three dimensions? That is, what is the 3D equivalent of two lines sharing the same "gradient" without intersecting?
In two dimensions, “gradient” (or slope) is just a single number telling you how much a line rises for a given run in the x–direction. In three dimensions you need to say how a line moves in all three directions (x,y,z), so instead of one number you use a three‐component object called a direction vector.
– If a line goes “over 2 in x, up 1 in y, and forward 3 in z,” we represent that by the vector ⟨2,1,3⟩. This simply records the proportions of movement in each coordinate.
– Two lines in space are parallel exactly when their direction vectors point the same way (or exactly opposite ways). In practice that means one vector is a scalar multiple of the other. For example, ⟨2,1,3⟩ and ⟨4,2,6⟩ both point in the same direction in 3D.
– Finally, to be truly distinct parallel lines, they must share no common point. You can picture them like train tracks floating in space—running side by side without ever meeting.
So the 3D equivalent of “same gradient but no intersection” is:
two lines whose direction vectors are proportional (same “3D slope”),
and whose points in space do not coincide.
Parallel lines in 3D
Two vector lines are parallel if their direction vectors are scalar multiples of each other and the lines are not the same.
Consider two lines:
These lines are parallel if b1=kb2 for some scalar k, but a1 does not lie on r2.
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Example
Consider two lines given by:
These lines are parallel because their direction vectors are parallel (one is a scalar multiple of the other), but the point (1,1,0) from the second line does not lie on the first line since
simultaneously implies that λ=0,21 and −21.
Checkpoint
Two lines are given by
Find the value of a for which r1 and r2 are parallel.
Select the correct option
The reason we put such an emphasis on parallel lines not sharing a point is because, just like in two dimensions, if they do share a point they are the same line. However it is often more difficult to tell if two lines share a point in 3D than it is in 2D due to the additional variable. For this reason, we have a special name for these kinds of three dimensional vector lines.
Coincident lines
Two vector lines are coincident if they represent exactly the same line, meaning every point on one line also lies on the other. For lines given by:
they are coincident if:
Their direction vectors are parallel, so b1=kb2.
A point from one line (e.g., a2) lies on the other line, satisfying a2=a1+λb1 for some scalar λ.
Example
Consider two lines given by:
These lines are coincident because their direction vectors are parallel (one is a scalar multiple of the other), and the point (3,6,9) from the second line lies on the first line (taking λ=1). Thus both lines represent exactly the same set of points.
Checkpoint
Two lines are given by
State whether the lines are parallel or coincident.
Select the correct option
Exercise
Two lines are given by
Find the value of a for which r1 and r2 are coincident.
Select the correct option
Just like in two dimensions, a pair of vector lines can also intersect if they share a single common point. We find this common point via the same process as in two dimensions: setting the equations equal and solving for the values of λ and μ.
As in 2D, 3D lines can either have exactly zero, one, or infinite points in common. Despite the extra dimension, they are still straight lines, and can never meet in, say, exactly two spots.
Intersecting lines
Two vector lines intersect if they share exactly one common point. That means they are not parallel.
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Example
Consider two lines given by
Setting them equal, ⎝⎛1+4λ23⎠⎞=⎝⎛2+3μ1+μ1+2μ⎠⎞, leads to the system
1+4λ=2+3μ, 2=1+μ, 3=1+2μ. Solving gives μ=1 and λ=1. Substituting back shows the unique intersection point is ⎝⎛523⎠⎞.
Exercise
Two lines are given by
Find the coordinates of their point of intersection P.
Select the correct option
In two dimensions, if two lines do not share at least one point of intersection, they must be parallel. In three dimensions, this is not necessarily the case.
Discussion
Consider two lines given by
First, show that the lines are not parallel or coincident.
Next, try to find a point where the two lines intersect. Is this possible? Why or why not?
How would you describe the relationship between these two lines? Why do you think this kind of relationship is possible in three dimensions but not two?
First, compare the direction vectors
Since v2 is not a scalar multiple of v1 (for instance, their third components differ), the lines are neither parallel nor coincident.
Next, to see if they meet, set
(from x,y,z–coordinates respectively). The first gives λ=μ. Substituting into the second yields
Because they are not parallel yet never meet, each line “passes by” the other in space without touching. You can picture this by thinking of two straight sticks in a room—one can run above or below the other so that they never cross. On a flat sheet (two dimensions), however, you have only “left–right” and “up–down,” so two straight lines that are not pointing the same way must cross somewhere. That extra “in–and–out” direction in three dimensions is what lets lines miss each other without ever touching.
We say vector lines that have different direction vectors and yet don't intersect are skew.
Skew lines
Two lines in three-dimensional space are skew if they are neither parallel nor intersecting.
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Example
Consider two lines given by
Their direction vectors ⎝⎛123⎠⎞ and ⎝⎛210⎠⎞ are not scalar multiples of each other, so the lines are not parallel. If you try to set r1=r2 to find an intersection, you get a system of equations that yields no consistent solution:
Setting r1=r2 yields the system:
Equating components:
From 3+3λ=0, we get λ=−1. Substituting into 1+λ=2μ gives 1−1=0=2μ⟹μ=0.
Then the second equation becomes 2+2(−1)=1+0⟹0=1, a direct contradiction.
Since no (λ,μ) pair satisfies all equations, the lines do not intersect, confirming they are skew.
Exercise
Two lines are given by
Show that r1 and r2 are skew.
Select the correct option
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