JEE Main & Advanced Mathematics Three Dimensional Geometry Shortest Distance Between two Straight Lines

(1) Skew lines : Two straight lines in space which are neither parallel nor intersecting are called skew lines.

Thus, the skew lines are those lines which do not lie in the same plane.

(2) Line of shortest distance : If \[{{l}_{1}}\] and \[{{l}_{2}}\] are two skew lines, then the straight line which is perpendicular to each of these two non-intersecting lines is called the “Line of shortest distance.”

There is one and only one line perpendicular to each of lines \[{{l}_{1}}\] and \[{{l}_{2}}\].

(3) Shortest distance between two skew lines

Let two skew lines be, \[\frac{x-{{x}_{1}}}{{{l}_{1}}}=\frac{y-{{y}_{1}}}{{{m}_{1}}}=\frac{z-{{z}_{1}}}{{{n}_{1}}}\] and \[\frac{x-{{x}_{2}}}{{{l}_{2}}}=\frac{y-{{y}_{2}}}{{{m}_{2}}}=\frac{z-{{z}_{2}}}{{{n}_{2}}}\]

Therefore, the shortest distance between the lines is given by  \[d=\frac{\left| \,\begin{matrix} {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}}  \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\ \end{matrix}\, \right|}{\sqrt{{{({{m}_{1}}{{n}_{2}}-{{m}_{2}}{{n}_{1}})}^{2}}+{{({{n}_{1}}{{l}_{2}}-{{l}_{1}}{{n}_{2}})}^{2}}+{{({{l}_{1}}{{m}_{2}}-{{m}_{1}}{{l}_{2}})}^{2}}}}\].