Skew Lines

Skew Lines: Skew lines refer to a pair of lines that neither intersect nor run parallel to each other. This concept only applies in spaces with more than two dimensions, as skew lines must reside in separate planes, making them non-coplanar. In contrast, within two-dimensional space, lines are limited to two relationships: they can either cross each other or be parallel.

In this article, we will learn about skew lines, examples of skew lines, and how to calculate the shortest path between skew lines and other details.

What are Skew Lines?

We must first learn about their types of lines before learning more about skew lines, which include:

• Intersecting Lines: Two or more lines are considered intersecting if they are in the same plane and intersect at a certain place.
• Parallel Lines: Two lines are considered parallel if they lay in the same plane and never cross, even when stretched indefinitely.
• Coplanar Lines: Coplanar Lines reside in the same plane.

Now about skew lines:

• Skew Lines: Non-parallel Lines and non-intersecting are called skew lines.

Skew Lines Definition

A pair of non-intersecting, non-parallel, and non-coplanar lines are known as skew lines. It follows from this that skew lines are not parallel to one another and can never cross.

Lines can be parallel or intersecting to exist in two dimensions or the same plane. Skewed lines will always be non-coplanar and exist in three or more dimensions since they are not subject to this characteristic.

Skew Lines Examples

A lot of real-world scenarios have skewed lines. Let’s say there are two lines: one on the ceiling and one on the wall. These lines may be skewed if they do not meet and are not parallel to one another since they are located in separate planes. These lines never end in either direction.

Skew Lines in 3D

Since skew lines are inherently non-coplanar, they will always exist in three dimensions.

Assume the following three-dimensional solid shape as shown in the image below. On the triangle face, we draw a single line that we call “a.” We designate ‘b’, the single line that we draw on the quadrilateral-shaped face.

There isn’t a plane that contains both a and b. “a” and “b” are not parallel to one another and will never intersect if we stretch them indefinitely in both directions. Thus, in 3D, “a” and “b” are skew lines.

Skew Lines Example

Skew Lines in a Cube

One example of a solid three-dimensional form is a cube. We take three steps in order to locate skew lines in a cube.

Step 1: Locate lines that don’t cross over.

Step 2: Determine whether or not these line pairs are likewise not parallel to one another.

Step 3: Next, determine if these lines are non-coplanar if they are not intersecting or parallel. If so, the selected pair of lines is skew.

The cube is given below as:

It is evident that lines GF and CD are neither parallel or intersecting. Moreover, their planes of lying are not the same. CD and GF are hence skew lines.

When looking for skew lines, one may also include the diagonals of solid objects.

Skew Lines Formula

Skewed lines don’t exist in two-dimensional space. We have formulae for calculating the shortest path between skew lines in three dimensions, utilizing both the vector and cartesian methods. Since the two skew lines are not parallel and never cross, it might be difficult to calculate the angle between them.

Angle Formed by Two Skew Lines

Assume PQ and RS, our two skew lines. Take point O on RS, and draw OT, a line parallel to PQ, from this point. The measurement of the angle between the two skew lines may be obtained using the angle SOT.

Formula for Distance Between Skew Lines

We must draw a line perpendicular to these two skew lines in order to determine the separation between them. To get various versions of the formula for the shortest distance between two selected skew lines, we can express these lines in both cartesian and vector forms.

Assume P₁ and P₂ are our two skew lines. In the following part, we will examine how to calculate the separation between two skew lines.

Vector Form

Vector form of P₁:

[Tex]\vec{l}_1 = \vec{m}_1 + t \cdot \vec{n}_1 [/Tex]

Vector form of P₂:

[Tex]\vec{l}_2 = \vec{m}_2 + t \cdot \vec{n}_2 [/Tex]

In this case, a point on line P₁ is E = −[Tex]\vec{m}_1[/Tex], while a point on line P₂ is F = [Tex]\vec{m}_2[/Tex].

From E to F, the vector is [Tex]\vec{m}_2[/Tex]– −[Tex]\vec{m}_1[/Tex]. The symbols [Tex]\vec{n}_1[/Tex] and [Tex]\vec{n}_2[/Tex] in this case denote the directions of lines P₁ and P₂, respectively. The actual number t is what establishes the point’s location on the line. The following is the unit normal vector to P₁ and P₂:

[Tex]\vec{n} = \frac{\vec{n}_1 \times \vec{n}_2}{|\vec{n}_1 \times \vec{n}_2|} [/Tex]

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

[Tex]d = \left| (\vec{n}_1 \times \vec{n}_2) \cdot (\vec{m}_2 – \vec{m}_1) \right| \div |\vec{n}_1 \times \vec{n}_2| [/Tex]

Cartesian Form

To find the shortest distance between lines P₁ and P₂, we shall take into account their symmetric equations.

[Tex] \frac{x – x_1}{a_1} = \frac{y – y_1}{b_1} = \frac{z – z_1}{c_1} [/Tex]

[Tex] \frac{x – x_2}{a_2} = \frac{y – y_2}{b_2} = \frac{z – z_2}{c_2} [/Tex]

where,

• a, b, and c represent Direction Vectors of Lines

Consequently, the following is the cartesian equation for the shortest path between skew lines:

[Tex]d = \left| \frac{{(x_2 – x_1)(y_2 – y_1)(z_2 – z_1)}}{{a_1b_1c_1a_2b_2c_2}} \right| \sqrt{(b_1c_2 – b_2c_1)^2 + (c_1a_2 – c_2a_1)^2 + (a_1b_2 – a_2b_1)^2} [/Tex]

Distance Between Skew Lines

Drawing a line perpendicular to both lines will reveal the distance between the skew lines. To calculate the distance, we may apply the previously given cartesian and vector formulae.

Distance Between Two Skew Lines

We may use any of the two distance formulae, depending on the kind of equations provided, to determine the separation between two skew lines. When determining the distance, we have two options: the symmetric equations or the parametric equations of a line.

Shortest Distance Between Two Skew Lines

Shortest distance between two skew lines is the perpendicular line between two skew lines. Line that is perpendicular to two skew lines is the shortest distance between them, not the line that connects both skew lines.

Given is the vector equation: [Tex]d = \left| (\vec{n}_1 \times \vec{n}_2) \cdot (\vec{m}_2 – \vec{m}_1) \right| / |\vec{n}_1 \times \vec{n}_2|[/Tex]is used to parametric equations to depict the lines.

Cartesian equation is [Tex]d = \left| \frac{{(x_2 – x_1)(y_2 – y_1)(z_2 – z_1)}}{{a_1b_1c_1a_2b_2c_2}} \right| \sqrt{(b_1c_2 – b_2c_1)^2 + (c_1a_2 – c_2a_1)^2 + (a_1b_2 – a_2b_1)^2} [/Tex] is applied when the symmetric equations represent the lines.

Notes on Skew Lines

• Skewed lines are those that are not coplanar, parallel, or intersecting.
• Only in three or more dimensions are skew lines possible. Therefore, skew lines are not possible in 2D space.
• It is possible to find the shortest distance between skew lines in both vector and cartesian form using this formula.

Skew Lines: Solved Examples

Example 1: Find the shortest between two lines:

l₁ = (i + 3j) + t(2i – j + k)

l₂ = (3i – 2j – 4k) + t(4i + 3k)

Solution:

Using vector form of equations,

m₁ = (i + 3j), n₁ = (2i – j + k)

m₂ = (3i – 2j – 4k), n₂ = (4i + 3k)

m₂ – m₁ = (2i – 5j – 4k)

n₁×n₂ = (-3i – 2j + 4k)

|n₁×n₂| = 5.385

Substituting in [Tex]d = \left| (\vec{n}_1 \times \vec{n}_2) \cdot (\vec{m}_2 – \vec{m}_1) \right| / |\vec{n}_1 \times \vec{n}_2|[/Tex]

d = |(-12)/5.385|

d = 2.228 units

Example 2: Find the shortest between two lines:

l₁ = (2i – j) + t(i + 2j + 3k)

l₂ = (i – j – k) + t(i + 3j – 2k)

Solution:

Using vector form of equations,

m₁ = (2i – j), n₁ = (i + 2j + 3k)

m₂ = (i – j – k), n₂ = (i + 3j – 2k)

m₂ – m₁ = (-i – k)

n₁×n₂ = (-13i + 5j + k)

|n₁×n₂| = 13.96

Substituting in [Tex]d = \left| (\vec{n}_1 \times \vec{n}_2) \cdot (\vec{m}_2 – \vec{m}_1) \right| / |\vec{n}_1 \times \vec{n}_2|[/Tex]

d = |(-14)/13.96|

d = 1.003 units

Practice Problems on Skew Lines

Problem 1: Determine if the following pairs of lines are skew or not:

• l₁​: x = 1 + t, y = 2 – t, z = 3t
• l₂​: x = 2 – 2t, y = -1+3t, z = 4 + 2t

Problem 2: Given a skew line l: x = 3 – t, y = 2t, z = 1 + 2t, and a point P(1, -2, 3), find the shortest distance between the line and the point.

FAQs on Skew Lines

What are skew lines in Mathematics?

Lines that are not parallel or intersect are known as skew lines. Their locations are distinct and they don’t intersect in any way.

How to find whether two lines are skewed?

By comparing the direction vectors of two lines, we may ascertain if they are skew. The lines are skew if the direction vectors are neither parallel nor antiparallel.

Are coplanar skew lines possible?

Skewed lines are not coplanar. Two lines cannot be skewed if they are in the same plane; they must either intersect or run parallel.

Are Parallel Lines Skew Lines?

No, parallel line are not skew lines.

Are skew lines equidistant?

No, skew lines are not equidistant.

Are skew lines coplanar?

No, skew lines are not coplanar.