While finding the distance between two lines, my book says that you will be asked for finding distance between skew lines or parallel lines, because intersecting lines would have this $0$.
But my question is how can I know whether the equation in front of me is of skew lines or intersecting lines ? (I can identify parallel lines)
Best Answer
Two lines are skew if and only if they are not coplanar (they are not in a same plane). So take two distinct points on the first line and two distinct points on the second line and check if those four points are coplanar: four points $P_i:=(x_i,y_i,z_i)$ for $i=1,2,3,4$ are coplanar if and only if the vectors $P_2-P_1$, $P_3-P_1$, $P_4-P_1$ are linearly dependent that is if and only if the matrix $$\begin{bmatrix}x_2-x_1 & x_3-x_1 & x_4-x_1\\ y_2-y_1 & y_3-y_1 & y_4-x_1\\ z_2-z_1 & z_3-z_1 & z_4-z_1\end{bmatrix}$$ has determinant zero.