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What are Skew Lines

Skew Lines are two different straight lines that are not lying on the same plane in two different directions and do not meet each other. In the figure, line AB is located on a different plane and line CD is located on a different plane.

Skew Lines


Skew Lines are two or more non-parallel lines or curves that do not intersect. In 2-dimensional Euclidean geometry, there are no skew lines. In 3 or higher dimensions, there are an infinite number of skew lines.



Skew lines do not lie in a plane. The angle between two skew lines is defined as the angles between any two lines parallel to them and passing through a point in space. If A and B are the direction vectors of two skew lines, then the cosine of the angle between them is given by,

SKEW

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Common Perpendicular of Skew Lines

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The common perpendicular of two skew lines is the line intersecting both of them at right angles. Any two skew lines have a unique common perpendicular. The equation of this common perpendicular to the lines r = r1+ at1 and r = r2- bt2 has the form

((r-r1),a,[a,b])=0,

((r-r2),b,[a,b])=0,

Distance between Skew Lines

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The distance between two skew lines is the length of the segment of their common perpendicular whose end points lie on the lines. To find the shortest distance between these lines, we need to find the point view of any one line. The perpendicular from that point view on the corresponding view of the other line gives the required distance.

The distance between two skew lines is given by

skew formulae

Simple example of skew lines:

ST and UV are skew lines in the figure given below

Example of Skew Lines

Skew Lines Examples

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Which of the following are skew lines?

Skew Lines Example

Choices:
A.
RY and YR
B.
PS and PQ
C.
QR and RY
D.
RY and PS
Correct Ans: D
Sol:
Step 1: Two nonparallel lines in space that do not intersect are called Skew lines.
Step 2: Among the pairs, the lines RY and PS are nonparallel and lie in different planes, and do not intersect each other.
Step 3: So, RY and PS are skew lines among the given pairs.

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