There are three possible types of relations that two different lines can have in a three-dimensional space. They can be
- parallel, when their direction vectors are parallel and the two lines never meet;
- meeting at a single point, when their direction vectors are not parallel and the two lines intersect;
- skew, which means that they never meet and are not parallel.
All three of these relations can be found in a cuboid. In the cuboid shown in the diagram below, edges and are parallel. Edges and intersect at a single point Edges and are skew, since they are not parallel and never meet.
Find all edges that are skew to in the cuboid shown below.
We should find all the lines that do not meet with and are not parallel to which are edges and Observe that any edge in a cuboid is skew to four other edges.
Find all edges that are skew to in the pentagonal prism shown below.
We should find all the edges that do not meet with and are not parallel to The edges that meet with are and The edge is parallel to Therefore all the edges except for these are skew to which are edges and
Determine the relation between the following two lines:
The direction vectors of the two lines are and Since their direction vectors are not parallel, the two lines either intersect at a single point or are skew to each other.
Now let's find out if the two lines meet. Equating the first equation to gives
so any point on the first line can be expressed as Plugging this into the second equation gives
Observe that there is no such real number that satisfies this equation. Then the two lines do not meet, so they are skew (because they are not parallel, either, as proved earlier).
The (shortest) distance between a pair of skew lines can be found by obtaining the length of the line segment that meets perpendicularly with both lines, which is in the figure below.
Find the distance between the following pair of skew lines:
Let be and be We should find the length of which is the line segment that meets perpendicularly with both and Equating the equation of with gives
Hence point can be expressed as for some real number Applying the same method for gives
Thus point can be expressed as for some real number Then we have
Now, let denote the direction vector of and be that of Then we have and Since should be perpendicular to both and it must be true that Hence we have
Solving the simultaneous equations (1) and (2) gives
Therefore and the distance between the two skew lines is