3.5 Distance between two skew 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 us assume the equation of two skew lines be $${l_1} \equiv \overrightarrow r = \overrightarrow {{a_1}} + \lambda \overrightarrow {{b_1}} $$ and $${l_2} \equiv \overrightarrow r = \overrightarrow {{a_2}} + \mu \overrightarrow {{b_2}} $$

and $\overrightarrow {PQ} $ is perpendicular to both ${l_1}$ and ${l_2}$ as shown in figure. Therefore, we can say that $\overrightarrow {PQ} $ is the shortest distance vector between ${l_1}$ and ${l_2}$. Also $\overrightarrow {PQ} $ is parallel to $\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} $.

Now, let us assume $\widehat n$ to be the unit vector along $\overrightarrow {PQ} $, which can be written as $$\widehat n = \frac{{\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} }}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}}$$

$$\begin{equation} \begin{aligned} \therefore \overrightarrow {PQ} = {\text{projection of }}\overrightarrow {AB} {\text{ on }}\overrightarrow {PQ} \\ \Rightarrow \overrightarrow {PQ} = \overrightarrow {AB} .\widehat n \\ {\text{ = }} \pm \left( {{a_2} - {a_1}} \right).\frac{{\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} }}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}} \\ {\text{ = }} \pm \frac{{\left( {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right).\left( {{a_2} - {a_1}} \right)}}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}} \\\end{aligned} \end{equation} $$

Hence, distance between two skew lines is $$PQ = \left| {\frac{{\left( {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right).\left( {{a_2} - {a_1}} \right)}}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}}} \right|$$

Now, if the equation of two skew lines are given in cartesian form i.e., $$\begin{equation} \begin{aligned} {l_1} \equiv \frac{{x - {x_1}}}{{{a_1}}} = \frac{{y - {y_1}}}{{{b_1}}} = \frac{{z - {z_1}}}{{{c_1}}} \\ {l_2} \equiv \frac{{x - {x_2}}}{{{a_2}}} = \frac{{y - {y_2}}}{{{b_2}}} = \frac{{z - {z_2}}}{{{c_2}}} \\\end{aligned} \end{equation} $$

Then the shortest distance between the lines is given by

\[\left| {\frac{{\begin{array}{c} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}}}{{\sqrt {{{\left( {{b_1}{c_2} - {b_2}{c_1}} \right)}^2} + {{\left( {{c_1}{a_2} - {c_2}{a_1}} \right)}^2} + {{\left( {{a_1}{b_2} - {a_2}{b_1}} \right)}^2}} }}} \right|\]