Skew Lines

Skew lines are a pair of lines that do not intersect and are not parallel to each other. Skew lines can only exist in dimensions higher than 2D space. They have to be non-coplanar meaning that such lines exist in different planes. In two-dimensional space, two lines can either be intersecting or parallel to each other. Thus, skew lines can never exist in 2D space.

Skew lines can be found in many real-life situations. Suppose there is a line on a wall and a line on the ceiling. If these lines are not parallel to each other and do not intersect then they can be skew lines as they lie in different planes. These lines continue in two directions infinitely. In this article, we will learn more about skew lines, their examples, and how to find the shortest distance between them.

Before learning about skew lines, we need to know three other types of lines. These are given as follows:

• Intersecting Lines - If two or more lines cross each other at a particular point and lie in the same plane then they are known as intersecting lines.
• Parallel Lines - If two are more lines never meet even when extended infinitely and lie in the same plane then they are called parallel lines.
• Coplanar Lines - Coplanar lines lie in the same plane.

Skew Lines Definition

Skew lines are a pair of lines that are non-intersecting, non-parallel, and non-coplanar. This implies that skew lines can never intersect and are not parallel to each other. For lines to exist in two dimensions or in the same plane, they can either be intersecting or parallel. As this property does not apply to skew lines, hence, they will always be non-coplanar and exist in three or more dimensions.

Skew Lines Example

In real life, we can have different types of roads such as highways and overpasses in a city. These roads are considered to be in different planes. Lines drawn on such roads will never intersect and are not parallel to each other thus, forming skew lines.

Skew lines will always exist in 3D space as these lines are necessarily non-coplanar. Suppose we have a three-dimensional solid shape as shown below. We draw one line on the triangular face and name it 'a'. We also draw one line on the quadrilateral-shaped face and call it 'b'. Both a and b are not contained in the same plane. If we extend 'a' and 'b' infinitely in both directions, they will never intersect and they are also not parallel to each other. Thus, 'a' and 'b' are examples of skew lines in 3D. In 3D space, if there is a slight deviation in parallel or intersecting lines it will most probably result in skew lines.

Skew Lines in a Cube

A cube is an example of a solid shape that exists in 3 dimensions. To find skew lines in a cube we go through three steps.

• Step 1: Find lines that do not intersect each other.
• Step 2: Check if these pairs of lines are also not parallel to each other.
• Step 3: Next, check if these non-intersecting and non-parallel lines are non-coplanar. If yes then the chosen pair of lines are skew lines.

Suppose we have a cube as given below:

We see that lines CD and GF are non-intersecting and non-parallel. Further, they do not lie in the same plane. Thus, CD and GF are skew lines.

Diagonals of solid shapes can also be included when searching for skew lines.

There are no skew lines in two-dimensional space. In three dimensions, we have formulas to find the shortest distance between skew lines using the vector method and the cartesian method. To determine the angle between two skew lines the process is a bit complex as these lines are not parallel and never intersect each other.

Angle Between Two Skew Lines

Suppose we have two skew lines PQ and RS. Take a point O on RS and draw a line from this point parallel to PQ named OT. The angle SOT will give the measure of the angle between the two skew lines.

Distance Between Skew Lines Formula

To find the distance between the two skew lines, we have to draw a line that is perpendicular to these two lines. We can represent these lines in the cartesian and vector form to get different forms of the formula for the shortest distance between two chosen skew lines.

Say we have two skew lines P1 and P2. We will study the methods to find the distance between two skew lines in the next section.

Vector Form

Vector form of P1: \(\overrightarrow{l_{1}} = \overrightarrow{m_{1}} + t.\overrightarrow{n_{1}}\)

Vector form of P2: \(\overrightarrow{l_{2}} = \overrightarrow{m_{2}} + t.\overrightarrow{n_{2}}\)

Here, E = \(\overrightarrow{m_{1}}\) is a point on the line P1 and F = \(\overrightarrow{m_{2}}\) is a point on P2. \(\overrightarrow{m_{2}}\) - \(\overrightarrow{m_{1}}\) is the vector from E to F. Here, \(\overrightarrow{n_{1}}\) and \(\overrightarrow{n_{2}}\) represent the direction of the lines P1 and P2 respectively. t is the value of the real number that determines the position of the point on the line. The unit normal vector to P1 and P2 is given as:

n = \(\frac{\overrightarrow{n_{1}}\times\overrightarrow{n_{2}}}{|\overrightarrow{n_{1}}\times\overrightarrow{n_{2}}|}\)

The shortest distance between P1 and P2 is the projection of EF on this normal. Thus, this is given by

d = |\(\frac{(\overrightarrow{n_{1}}\times\overrightarrow{n_{2}})(\overrightarrow{m_{2}}-\overrightarrow{m_{1}})}{|\overrightarrow{n_{1}}\times\overrightarrow{n_{2}}|}\)|

Cartesian Form

We will consider the symmetric equations of lines P1 and P2 to get the shortest distance between them.

Equation of P1: \(\frac{x - x_{1}}{a_{1}}\) = \(\frac{y - y_{1}}{b_{1}}\) = \(\frac{z - z_{1}}{c_{1}}\)

Equation of P2: \(\frac{x - x_{2}}{a_{2}}\) = \(\frac{y - y_{2}}{b_{2}}\) = \(\frac{z - z_{2}}{c_{2}}\)

here, a, b and c are the direction vectors of the lines.

Thus, the cartesian equation of the shortest distance between skew lines is given as

d = \(\frac{\begin{vmatrix} x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1}\\ a_{1}& b_{1} & c_{1}\\ a_{2}& b_{2} & c_{2} \end{vmatrix}}{[(b_{1}c_{2} - b_{2}c_{1})^{2}(c_{1}a_{2} - c_{2}a_{1})^{2}(a_{1}b_{2} - a_{2}b_{1})^{2}]^{1/2}}\)

The distance between skew lines can be determined by drawing a line perpendicular to both lines. We can use the aforementioned vector and cartesian formulas to find the distance.

Distance Between Two Skew Lines

Depending on the type of equations given we can apply any of the two distance formulas to find the distance between two lines which are skew lines. We can either use the parametric equations of a line or the symmetric equations to find the distance.

Shortest Distance Between Two Skew Lines

The shortest distance between two skew lines is given by the line that is perpendicular to the two lines as opposed to any line joining both the skew lines.

The vector equation is given by d = |\(\frac{(\overrightarrow{n_{1}}\times\overrightarrow{n_{2}})(\overrightarrow{a_{2}}-\overrightarrow{a_{1}})}{|\overrightarrow{n_{1}}\times\overrightarrow{n_{2}}|}\)| is used when the lines are represented by parametric equations

The cartesian equation is d = \(\frac{\begin{vmatrix} x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1}\\ a_{1}& b_{1} & c_{1}\\ a_{2}& b_{2} & c_{2} \end{vmatrix}}{[(b_{1}c_{2} - b_{2}c_{1})^{2}(c_{1}a_{2} - c_{2}a_{1})^{2}(a_{1}b_{2} - a_{2}b_{1})^{2}]^{1/2}}\) is used when the lines are denoted by the symmetric equations.

Related Articles:

• Vector Cross Product
• Plane Definition
• 3D Shapes

Important Notes on Skew Lines

• Lines that are non-intersecting, non-parallel, and non-coplanar are skew lines.
• Skew lines can only exist in three or more dimensions. Thus, we cannot have skew lines in 2D space.
• The formula to calculate the shortest distance between skew lines can be given in both vector form and cartesian form.

What are Skew Lines with Examples?

In three-dimensional space, if there are two straight lines that are non-parallel and non-intersecting as well as lie in different planes, they form skew lines. An example is a pavement in front of a house that runs along its length and a diagonal on the roof of the same house.

Are Parallel Lines Skew Lines?

According to the definition skew lines cannot be parallel, intersecting, or coplanar. Thus, parallel lines are not skew lines.

Are Skew Lines Equidistant?

As skew lines are not parallel to each other hence, even though they do not intersect at any point, they will not be equidistant to each other.

Are Skew Lines Non-Coplanar?

Lines that lie in the same plane can either be parallel to each other or intersect at a point. Thus, for two lines to be classified as skew lines, they need to be non-intersecting and non-parallel. As a consequence, skew lines are always non-coplanar.

How Do You Find a Skew Line?

We first check if the given lines lie in 3D space. Next, we check if they are parallel to each other. If they are not parallel we determine if these two lines intersect at any given point. If they do not intersect then such lines are skew lines.

What are Skew Lines in a Cube?

A cube is a 3D solid figure and hence, can have multiple skew lines. Skew lines in a cube can lie on any face or any edge of the cube as long as they do not intersect, are not parallel to each other, and do not lie in the same plane.

How Are Parallel Lines and Skew Lines Similar?

Parallel lines and skew lines are not similar. Parallel lines lie in the same plane and are equidistant to each other. However, skew lines are non-parallel, non-intersecting and thus, are non-coplanar.