Finding the Shortest Distance between two Skew Lines
In my previous post on the topic, i clarified that the shortest path between a pair of skew lines is the line segment that is perpendicular to both the skew lines. Now the next ask is determining a way to find the length of the shortest path, i.e. the length of the line segment perpendicular to both the skew lines. Let’s take an example of a pair of skew lines. L1 : \(\hat{i} + \hat{j} + 0\hat{k} + u(2\hat{i} - \hat{j} + \hat{k})\) L2 : \(2\hat{i} + \hat{j} - \hat{k} + v(3\hat{i} - 5\hat{j} + 2\hat{k})\) From equation of L1, position and direction vectors of L1 are \(\overrightarrow{p_1} = \hat{i} + \hat{j} + 0\hat{k}\) \(\overrightarrow{d_1} = 2\hat{i} - \hat{j} + \hat{k}\) From equation of L2, position and direction vectors of L2 are \(\overrightarrow{p_2} = 2\hat{i} + \hat{j} - \hat{k}\) \(\overrightarrow{d_2} = 3\hat{i} - 5\hat{j} + 2\hat{k}\) Direction vector of a line is simply a vector arrow parallel to that line. Or, we can also say it is a vector arrow ...