Coplanarity Two Lines – Ways to Identify Coplanar Lines and Solved Examples

Coplanarity of Two Lines In 3D Geometry

Coplanarity of two lines in 3D geometry is defined as the condition when the two lines lie in the same plane. In other words, the two lines are parallel to each other.

How do we Identify Coplanar Lines?

In mathematics, coplanar lines are lines that intersect in a single plane. To identify coplanar lines, we can use a number of different methods, including using a ruler and straightedge to draw lines on a piece of paper, using a compass to draw circles, or using a software program to draw lines on a computer screen.

What is the Condition of Vectors Coplanarity?

The condition of vectors coplanarity is when two or more vectors are aligned in the same plane. This can be done by using a vector diagram to determine if the vectors are collinear or not. If the vectors are collinear, then they will be in the same line and will have the same magnitude and direction. If the vectors are not collinear, then they will be in different planes and will have different magnitudes and directions.

Coplanarity of Lines Using Condition in Vector Form

The condition for lines to be coplanar is that the dot product of their vectors be zero.

This condition can be written in vector form as

Coplanarity of Lines Using Condition in Vector Form

The condition for lines to be coplanar is that their vectors be parallel. If two vectors are parallel, then they will have the same magnitude and direction. This condition can be written using vector form as:

|A| = |B| = magnitude of vectors A and B
direction of vectors A and B are parallel

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 sol. 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.