Coplanar Lines

Coplanarity of Two Lines

Coplanar lines are parallel or intersecting lines sharing the same plane. It is used to determine coplanarity conditions for two lines having the vector and Cartesian forms. This lesson also includes a solved example to ensure you can better understand the concepts discussed in this topic. Consider the following terms that are essential while learning about coplanar lines:

A line is a collection of points on both sides that extends endlessly.

• Coplanar lines: Two or more points or lines are said to be coplanar if they are positioned on the same plane
• Non Coplanar lines: When two or more points or lines do not lie on the same plane, they are said to be noncoplanar

Three-Dimensional Space: Coplanarity of Two Lines

Coplanar lines are used to operate in three-dimensional space, which is why they are called coplanar lines. The three-Dimensional form is represented by the box because there is only one plane in two-dimensional space. Everything that is two-dimensional in nature will be coplanar because there is only one plane in two-dimensional space. Consider the piece of paper in question. Whatever you draw on the piece of paper will be two-dimensional, and everything on it will be coplanar since everything is linked by the flat piece of paper that serves as a surface for drawing.

To determine if two lines are coplanar, we must examine the three-dimensional space; otherwise, there is nothing to examine or check. The only way to have more than one plane is in the three-dimensional world. Planes may be parallel to one another, or they can intersect with one another in various ways. 

Each of the box’s surfaces is a plane, and the diagonals are likewise planes. Consider the following scenario: you take a knife and cut the box diagonally. The surface formed as a consequence is likewise a plane. If you slice the box into slices like sliced bread, you will end up with a variety of planes or flat surfaces as a result of your slicing technique.

Some Examples of Coplanar Lines in the Real World

• On a notepad, the lines are parallel to one another and coplanar. They are on the same page because they are on the same plane. Unusual fact: these lines are not only parallel but also coplanar
• Our watches and clocks also include coplanar hands. The second, minute and hour hands are all grouped together in a circular space
• Grids found on graphing paper. Due to the fact that the grid’s vertical and horizontal lines are positioned on the same sheet of paper, they are referred to as coplanar grid positions

The Theoretical Concept of Coplanarity

Coplanar lines are a frequently discussed topic in three-dimensional geometry. Coplanarity is defined mathematically as the condition in which a certain number of lines all lie on the same plane. When this condition exists, the lines are said to be coplanar.

To refresh your recollection, a plane is a two-dimensional figure that extends into infinity in three-dimensional space. We depicted straight lines in this chapter using vector equations (also referred to as lines). Now we will examine the requirements or conditions that must exist for two lines to be coplanar.

Consider these are following equations for two straight lines when represented in vector form:

r1 = L1 + λQ­1

r2 = L2 + λQ2

What do these equations imply?

Let’s look at it this way. The first line intersects a point, let us call it L, whose position vector is given by L1 and is parallel to the Q1. Additionally, the second line is said to pass through another location whose vector is given by L2 and is parallel to Q2. This location is referred to as the second point.

Coplanarity requires that the line linking the two points is perpendicular to the product of the two vectors, Q1 and Q2, and parallel to the line connecting the two locations. We know that the vector form of the line connecting the two points is (L2 –L1). As a consequence, we have the following:

(L2 – L1). (Q1 x Q2) = 0

Coplanarity in Cartesian Form 

The vector form is used to define the criteria for coplanarity in the Cartesian form. Consider two places on the Cartesian plane denoted by the letters L and M: L (x1, y1, z1) and M (x2, y2, z2). Consider two vectors: Q1 and Q2. A1, B1, C1 and A2, B2, C2 are the first and second variables’ direction ratios, respectively.

We will now take the vector form of the condition stated before and convert it to the Cartesian form. This may be used for calculation in its present state. The two lines are coplanar if LM.(Q1 x Q2) = 0. As a result, the matrix corresponding to this equation in Cartesian form is indicated by 0.