3.3 Co-planarity of two lines

2.1 Vectorial form of a line passing through a given point and parallel to a given vector
2.2 Vectorial form of a line passing through two given points
2.3 Cartesian form of a line passing through a given point and parallel to a given vector
2.4 Cartesian form of a line passing through two given points

3.1 Condition for perpendicularity
3.2 Condition for parallelism
3.3 Co-planarity of two lines
3.4 Shortest distance between two lines
3.5 Distance between two skew lines
3.6 Distance between parallel lines
3.7 Perpendicular distance of a point from a line
3.8 Image of a point in a straight line

4.1 Equation of plane in normal form
4.2 Equation of a plane perpendicular to a given vector and passing through a given point
4.3 Equation of plane passing through a given point and parallel to the two given vectors
4.4 Equation of plane passing through three non-collinear points
4.5 Intercept form of plane
4.6 Equation of plane passing through the intersection of two given planes

5.1 Angle between two planes
5.2 Angle between a line and a plane
5.3 Distance of a point from a plane
5.4 Distance between two parallel planes

Let the vector equation of two lines be

$$\begin{equation} \begin{aligned} \overrightarrow r = \overrightarrow {{a_1}} + \lambda \overrightarrow {{b_1}} \;...(1) \\ \overrightarrow r = \overrightarrow {{a_2}} + \mu \overrightarrow {{b_2}} \;...(2) \\\end{aligned} \end{equation} $$

where $\overrightarrow {{a_1}} $ is the position vector of point $A$ through which line $(1)$ passes and parallel to $\overrightarrow {{b_1}} $ and $\overrightarrow {{a_2}} $ is the position vector of point $B$ through which line $(2)$ passes and parallel to $\overrightarrow {{b_2}} $. Therefore, we can write $$\overrightarrow {AB} = \overrightarrow {{a_2}} - \overrightarrow {{a_1}} $$

The given two lines are coplanar means in the same plane if and only if $\overrightarrow {AB} $ is perpendicular to $\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} $ i.e., $$\begin{equation} \begin{aligned} \overrightarrow {AB} .\left( {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right) = 0 \\ \left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right).\left( {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right) = 0 \\\end{aligned} \end{equation} $$

If the equation of two lines are given in cartesian form i.e., $$\begin{equation} \begin{aligned} \frac{{x - {x_1}}}{{{a_1}}} = \frac{{y - {y_1}}}{{{b_1}}} = \frac{{z - {z_1}}}{{{c_1}}} \\ \frac{{x - {x_2}}}{{{a_2}}} = \frac{{y - {y_2}}}{{{b_2}}} = \frac{{z - {z_2}}}{{{c_2}}} \\\end{aligned} \end{equation} $$

The given two lines are coplanar if and only if $\overrightarrow {AB} $ is perpendicular to $\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} $ where $$\begin{equation} \begin{aligned} \overrightarrow {AB} = \left( {{x_2} - {x_1}} \right)\widehat i + \left( {{y_2} - {y_1}} \right)\widehat j + \left( {{z_2} - {z_1}} \right)\widehat k \\ \overrightarrow {{b_1}} = {a_1}\widehat i + {b_1}\widehat j + {c_1}\widehat k \\ \overrightarrow {{b_2}} = {a_2}\widehat i + {b_2}\widehat j + {c_2}\widehat k \\\end{aligned} \end{equation} $$

Put all the values in the above equation, we get the condition for co-planarity of two lines as

\[\left| {\begin{array}{c} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}} \right| = 0\]