Coplanar

There are two words in geometry that start with "co" and sound similar and confusing. They are collinear and coplanar. In each of these words, "co" means together, "linear" means lying on a line, and "planar" means lying on a plane. Thus, collinear means that together lie on a line and coplanar means that together lie on a plane.

Let us learn more about coplanar points and coplanar lines in this article along with a few examples. Also, let us see how to determine whether given points are given lines are coplanar in coordinate geometry.

The word "coplanar" means "lying on the same plane". So obviously, "noncoplanar" means "don't lie on the same plane". In geometry, we study about two things with respect to coplanarity:

• Coplanar points
• Coplanar lines

The points that lie on the same plane are called coplanar points and hence the points that do NOT lie on the same plane are called non-coplanar points. We know that two points in 2D can always pass through a line and hence any two points are collinear. In the same way, three points in 3D can always pass through a plane and hence any 3 points are always coplanar. But four or more points in 3D may not be coplanar. So we define the coplanar points and non-coplanar points as follows with respect to the following example:

Coplanar Points Definition in Geometry

Four or more points that lie on the same plane are known as coplanar points. Remember that given any two points are always coplanar and given any three points are always coplanar. Here are some coplanar points examples from the above figure:

• A, B, C, and D are coplanar points.
• But each of F and E are NOT coplanar with A, B, C, and D.
• If any 3 points are taken at a time, a plane can pass through all those 3 points, and hence they are coplanar. For example:
  A, B, and E are coplanar.
  C, D, and F are coplanar.
  A, B, and E are coplanar, etc.

Non Coplanar Points Definition in Geometry

Two points are never non-coplanar and three points are also never non-coplanar. But four or more points are non-coplanar if they don't lie on a plane. For example, in the above figure, A, B, E, and F are non-coplanar points.

There are several methods to determine whether any 4 given points are coplanar. Let us learn each method. Consider the following example in each of the methods.

Example: Determine whether the four points A(1, -1, 2), B(3, -2, 5), C(1, 1, 4), and D(4, -2, 7) are coplanar.

Method 1 for Determining Coplanar Points

For any four points to be coplanar, find the equation of the plane through any of the three points and see whether the fourth point satisfies it.

Let us first find the equation of the plane through the first three points:
\((x_1,y_1,z_1)\) = (1, -1, 2)
\((x_2,y_2,z_2)\) = (3, -2, 5)
\((x_3,y_3,z_3)\) = (1, 1, 4)

For this, we use the equation of the plane formula:

\(\left|\begin{array}{ccc}
x-x_{1} & y-y_{1} & z-z_{1} \\
x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\
x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1}
\end{array}\right|=0\)

\(\left|\begin{array}{ccc}
x-1 & y+1 & z-2 \\
3-1 & -2+1 & 5-2 \\
1-1 & 1+1 & 4-2 \\
\end{array}\right|=0\)

\(\left|\begin{array}{ccc}
x-1 & y+1 & z-2 \\
2 & -1 & 3 \\
0 & 2 & 2 \\
\end{array}\right|=0\)

(x - 1) (-2 - 6) - (y + 1) (4 - 0) + (z - 2) (4 - 0) = 0
(x - 1) (-8) - (y + 1) (4) + (z - 2) (4) = 0
-8x + 8 - 4y - 4 + 4z - 8 = 0
-8x - 4y + 4z - 4 = 0
Divide both sides by -4,
2x + y - z + 1 = 0

Now, we will substitute the fourth point (x, y, z) = (4, -2, 7) in it and see whether it is satisfied.
2(4) + (-2) - 7 + 1 = 0
8 - 2 - 7 + 1 = 0
0 = 0, it satisfied.

Therefore, the given points are coplanar.

Method 2 for Determining Coplanar Points

For any given four points A, B, C, and D, find 3 vectors, say \(\overrightarrow{A B}\), \(\overrightarrow{BC}\), and \(\overrightarrow{CD}\) to be coplanar see whether their scalar triple product (determinant formed by the vectors) is 0.

Let us find the vectors \(\overrightarrow{A B}\), \(\overrightarrow{BC}\), and \(\overrightarrow{CD}\).

• \(\overrightarrow{A B}\) = B - A = (3, -2, 5) - (1, -1, 2) = (2, -1, 3)
• \(\overrightarrow{BC}\)= C - B = (1, 1, 4) - (3, -2, 5) = (-2, 3, -1)
• \(\overrightarrow{CD}\) = D - C = (4, -2, 7) - (1, 1, 4) = (3, -3, 3)

Now, their scalar triple product is nothing but the determinant formed by these three vectors. Let us find it and see whether it is 0.

\(\left|\begin{array}{ccc}
2 & -1 & 3 \\
-2 & 3 & -1 \\
3 & -3 & 3 \\
\end{array}\right|\)

= 2 (9 - 3) + 1 (-6 + 3) + 3 (6 - 9)
= 12 - 3 - 9
= 0

Therefore, the given four points are coplanar.

Method 3 for Determining Coplanar Points

For any given 4 points \((x_1, y_1,z_1)\), \((x_2,y_2,z_2)\), \((x_3,y_3,z_3)\), and \((x_4,y_4,z_4)\) to be coplanar, see if the 4x4 determinant \(\left|\begin{array}{llll}
x_{1} & y_{1} & z_{1} & 1 \\
x_{2} & y_{2} & z_{2} & 1 \\
x_{3} & y_{3} & z_{3} & 1 \\
x_{4} & y_{4} & z_{4} & 1
\end{array}\right|\) is 0.

But this process may be difficult because calculating the 4x4 determinant is difficult.

Two or more lines are said to be coplanar if they lie on the same plane, and the lines that do not lie in the same plane are called non-coplanar lines. Consider the following rectangular prism.

Coplanar Lines in Geometry

In the above rectangular prism, here are some coplanar lines:

• AD and DH as they lie on the left side face of the prism (i.e., on the same plane).
• AB and CD as they lie on the bottom face of the prism (i.e., on the same plane).
• BC and FG as they lie on the right side face of the prism (i.e., on the same plane).

Non-Coplanar Lines in Geometry

In the above rectangular prism, the following are some non-coplanar lines as they don't lie on the same plane (i.e., they don't lie on the same rectangle in this case).

• AD and GH
• AB and CG
• BC and EH

Two lines are said to be coplanar if they are present in the same plane. Here are the conditions for two lines to be coplanar both in vector form and cartesian form.

Condition For Coplanarity of Lines in Vector Form

If the vector equations of two lines are of the form \(\overrightarrow{r}\) = \(\overrightarrow{a}\) + k \(\overrightarrow{p}\) and \(\overrightarrow{r}\) = \(\overrightarrow{b}\) + k \(\overrightarrow{q}\) then they are coplanar if and only if \((\overrightarrow{b} - \overrightarrow{a}) \cdot (\overrightarrow{p} \times \overrightarrow{q})\) = 0.

Condition For Coplanarity of Lines in Cartesian Form

If the cartesian equations of two lines are of the form \(\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}\) and \(\frac{x-x_2}{a_2}=\frac{y-y_2}{b_2}=\frac{z-z_2}{c_2}\) then the lines are coplanar if and only if the determinant \(\left|\begin{array}{ccc}
x_2-x_1 &y_2-y_1 & z_2-z_1 \\
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
\end{array}\right|\) = 0.

• Any two points are always coplanar.
• Any three points are always coplanar.
• Four or more points are coplanar if they all are present on one plane.
• Two or more lines are coplanar if they all are present on one plane.

☛ Related Topics:

• Determinant Calculator
• Collinearity
• Collinear Vectors

What is Coplanarity Meaning?

"Coplanarity" means "being coplanar". In geometry, "coplanar" means "lying on the same plane". Points that lie on the same plane are coplanar points whereas lines that lie on the same plane are coplanar lines.

What is Coplanar Definition Geometry?

"Coplanar" is derived from two words.

• "co" - means "together"
• "planar" - means "lying on a plane"

So "coplanar" means "lying on the same plane".

What are Coplanar Examples?

Here are some examples of coplanar points and coplanar lines:

• Any two lines (edges) that lie on the same face of a cube are coplanar lines.
• The points that lie on a blackboard are coplanar points.

What is Non-Coplanar Definition Geometry?

"Coplanar" means "being lying on the same plane" and hence "non-coplanar" means "not being lying on the same plane".

If Three Points are Coplanar They Are Collinear. Is this True?

Assume that there are three points on a paper but they are NOT passing through a single line. In that case, they are coplanar but not collinear. So if three points are coplanar they don't need to be collinear. So the given statement is false.

How to Determine Whether Given 4 Points are Coplanar?

To find whether any given 4 points are coplanar, just see whether the scalar triple product of any 3 non-collinear vectors formed by the 4 points is 0. For example for any given 4 points P, Q, R, and S, we can see whether [PQ QR RS] = 0.

How to Determine Whether given 2 Lines are Coplanar?

The following are the conditions to determine whether 2 lines are coplanar.

• Vector form:
  Two lines \(\overrightarrow{r}\) = \(\overrightarrow{a}\) + k \(\overrightarrow{p}\) and \(\overrightarrow{r}\) = \(\overrightarrow{b}\) + k \(\overrightarrow{q}\) are coplanar if and only if \((\overrightarrow{b} - \overrightarrow{a}) \cdot (\overrightarrow{p} \times \overrightarrow{q})\) = 0.
• Cartesian form:
  \(\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}\) and \(\frac{x-x_2}{a_2}=\frac{y-y_2}{b_2}=\frac{z-z_2}{c_2}\) are coplanar if and only if the determinant \(\left|\begin{array}{ccc}
  x_2-x_1 &y_2-y_1 & z_2-z_1 \\
  a_1 & b_1 & c_1 \\
  a_2 & b_2 & c_2 \\
  \end{array}\right|\) = 0.