Coplanarity of Two Lines In 3D Geometry

Two lines are said to be coplanar when they both lie on the same plane in a three-dimensional space. We have learnt how to represent the equation of a line in three-dimensional space using vector notations. In this article, we will learn about coplanarity of two lines.

Condition for coplanarity of two lines in vector form:

Using vector notations equation of line is given by:

\(\vec{r}\) = \(\vec{l_{1}}\) + λ\(\vec{m_{1}}\)       ——————— (1)

\(\vec{r}\) = \(\vec{l_{2}}\) + μ\(\vec{m_{2}}\)  ——————– (2)

Here, the line (1) passes through a point L having position vector \(\vec{l_{1}}\) and is parallel to \(\vec{m_{1}}\) and the line (2) passes through a point M having position vector \(\vec{l_{2}}\) and is parallel to \(\vec{m_{2}}\). These two lines are coplanar if and only if \(\vec{LM}\) is perpendicular to \(\vec{m_{1}}\) x \(\vec{m_{2}}\).

This can be given as:

\(\vec{LM}\) = \(\vec{l_{2}}\)\(\vec{l_{1}}\)

Thus condition of coplanarity is given by:

\(\vec{LM}\).(\(\vec{m_{1}}\) x \(\vec{m_{2}}\)) = 0
(\(\vec{l_{2}}\)\(\vec{l_{1}}\)).(\(\vec{m_{1}}\) x \(\vec{m_{2}}\)) = 0

Condition for coplanarity of two lines in cartesian form:

Let us take two points L and M such that (x1, y1, z1) and (x2, y2, z2) be the coordinates of the points respectively. The direction cosines of two vectors \(\vec{m_{1}}\) and \(\vec{m_{2}}\) is given by a1, b1, c1 and a2, b2, c2 respectively.

\(\vec{LM}\) = (x2 − x1)\(\hat{i}\) + (y2 − y1)\(\hat{j}\) + (z2 − z1)\(\hat{k}\)

\(\vec{m_{1}}\) = a1\(\hat{i}\)  +b1\(\hat{j}\)   + c1\(\hat{k}\)

\(\vec{m_{2}}\) = a2\(\hat{i}\)  +b2\(\hat{j}\)  + c2\(\hat{k}\)

By the above condition two lines can be coplanar if and only if,

\(\vec{LM}\).(\(\vec{m_{1}}\) x \(\vec{m_{2}}\)) = 0

This can be represented in cartesian form as,

Cartesian Form

Problems related to coplanarity of two lines:

Question: Show that lines \(\frac{x + 3}{-3}\) = \(\frac{y – 2}{4}\) = \(\frac{z – 5}{5}\) and \(\frac{x + 1}{-3}\) = \(\frac{y – 2}{3}\) = \(\frac{z + 5}{6}\) are coplanar.


According to the question:

x1 = -3, y1 = 2, z1 = 5, x2 = -1 , y2 = 2, z2 = -5, a1 = -3 , b1 = 4, c1 = 5, a2 = -3 , b2 = 3, c2 = 6

Coplanarity of Two Lines

Thus, the given lines are not coplanar. To learn more about coplanarity of two lines download Byju’s- The Learning App.

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