Three Dimensional Geometry

Coplanarity of Two Lines

Coplanar lines are the lines that lie on the same plane. Prove that two lines are coplanar using the condition in vector form and Cartesian form. This lesson provides you with a solved example for you to assess your understanding based on the concepts here.

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Coplanarity in Theory

Coplanar lines are a common topic in three-dimensional geometry. In mathematical theory, we may define coplanarity as the condition where a given number of lines lie on the same plane, they are said to be coplanar.

To recall, a plane is a two-dimensional figure extending into infinity in the three-dimensional space, while we have used vector equations to represent straight lines (also referred to as lines). We shall now take a look at what condition is necessary to be fulfilled for two lines to be coplanar.

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Coplanarity of Two Lines

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Source: MathCaptain

Condition for Coplanarity in Vector Form

In vector form, let us consider the equations of two straight lines to be as under:

  • r1 = l1 + λm­1
  • r2 = l2 + λm2

What do these equations mean? It means that the first line passes through a point, say L, whose position vector is given by l1 and is parallel to m1. Similarly, the second line is said to pass through another point whose position vector is given by l2 and is parallel to m2.

The condition for coplanarity is that the line joining the two points must be perpendicular to the product of the two vectors, m1 and m2. To illustrate this, we know that the line joining the two said points can be written in vector form as (l2 – l1). So, we have:

(l2 – l1) . (m1 x m2) = 0

Condition for Coplanarity in Cartesian Form

The derivation of the condition for coplanarity in Cartesian form stems from the vector form. Let us consider two points L (x1, y1, z1) & M (x2, y2, z2) in the Cartesian plane. Let there be two vectors m1 and m2. Their direction ratios are given by a1, b1, c1 and a2, b2, c2 respectively.

The vector equation of the line joining L and M can be given as:

LM = (x2 – x1)i + (y2 – y1)j + (z2– z1)k

m1 = a1i + b1j + c1k

m2 = a2i + b2j + c2k

We shall now use the above condition in vector form to derive our condition in Cartesian form. This can be used as is for calculation purposes. By the condition above, the two lines would be coplanar if  LM. (m1 x m2) = 0. Therefore, in Cartesian form, the matrix representing this equation is given as 0.

Given below is a solved problem on how to prove that two lines are coplanar.

Solved Example for You on Coplanar Lines

Question 1: Are the lines (x + 3)/3 = (y – 1)/1 = (z – 5)/5 and (x + 1)/ -1 = (y – 2)/2 = (z – 5)/5 coplanar?

Answer: Comparing the equations with the general form, we have:

(x1, y1, z1) = (-3, 1, 5) and (x2, y2, z2) = (-1, 2, 5).

Note that a1, b1, c1 = -3, 1, 5 and a2, b2, c2 = -1, 2, 5.

So, by Cartesian form, we must solve the matrix:

= 2(5 – 10) – 1(-15 + 5) + 0(-6 + 1) = -10 + 10 = 0

Since the solution of the matrix gives us 0, we can say that the given lines are coplanar

Question 2: What is meant by coplanar?

Answer: Coplanar points refer to three or more points which all exist in the same plane. Any set of three points in space is said to be coplanar. A set of four points may be coplanar or it may not be coplanar.

Question 3: Can we say that collinear points are coplanar?

Answer: Collinear points are those whose existence takes place in the same line. Coplanar points are points that are all in the same plane. So, in case of collinear points, a person can choose one of infinite number of planes which has the line on which these points exist. As such, one can say that they are coplanar.

Question 4: How can one prove that two vectors are coplanar?

Answer: One can prove that two vectors are coplanar if they are in accordance with the following conditions:

  • In case the scalar triple product of any three vectors happens to be zero.
  • If any three vectors are such that they are linearly dependent
  • n vectors will be coplanar if among them no more than two vectors exist that are linearly independent vectors.

Question 5: Two points determine how many lines?

Answer: Two points determine only one line.

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