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:

\(\vec{r}\)

\(\vec{r}\)

Here, the line (1) passes through a point L having position vector \(\vec{l_{1}}\)

This can be given as:

\(\vec{LM}\)

Thus condition of coplanarity is given by:

\(\vec{LM}\)

(\(\vec{l_{2}}\)

#### Condition for coplanarity of two lines in cartesian form:

Let us take two points L and M such that (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) be the coordinates of the points respectively. The direction cosines of two vectors \(\vec{m_{1}}\)_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} respectively.

\(\vec{LM}\)_{2} − x_{1})\(\hat{i}\)_{2} − y_{1})\(\hat{j}\)_{2} − z_{1})\(\hat{k}\)

\(\vec{m_{1}}\)_{1}\(\hat{i}\)_{1}\(\hat{j}\)_{1}\(\hat{k}\)

\(\vec{m_{2}}\)_{2}\(\hat{i}\)_{2}\(\hat{j}\)_{2}\(\hat{k}\)

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

\(\vec{LM}\)

This can be represented in cartesian form as,

#### Problems related to coplanarity of two lines:

__Question:__ Show that lines \(\frac{x + 3}{-3}\)

__Solution: __

According to the question:

x_{1} = -3, y_{1} = 2, z_{1} = 5, x_{2} = -1 , y_{2} = 2, z_{2} = -5, a_{1} = -3 , b_{1} = 4, c_{1} = 5, a_{2} = -3 , b_{2} = 3, c_{2} = 6

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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