The Perpendicular Distance between two Skew Lines

Problem: Find the perpendicular distance between the line passing through the the point (1, -1, 1) which is parallel to the vector u =[1, 3, 0] and the line passing through the point (1, 1, 3) which is parallel to the vector v = [1, 1, 0].
A diagram of this is shown on the right.		A is the given point through which the first line passes .B is the given point through which the second line passes. MN is the common perpendicular to both given lines. u and v are the vectors to which the respective lines are parallel.
A line is drawn joining the points A and B.		Since MN is the common perpendicular, the length MN is the required length.
Since points A and B are on lines perpendicular to MN, MN = the projection of AB onto MN; Therefore, MN = AB .	AB = [1, 1, 3] - [1, -1, 1] = [0, 2, 2] Now, the cross product of two vectors gives a third vector which is perpendicular to both vectors. MN is perpendicular to both u and v, Therefore, MN will beparallel to u × v, but we can not be sure whether u × v is directed M to N or N to M Therefore, = +/- u × v = [1, 3, 0] × [1, 1, 0] = = i - j + k = [0, 0, -2] \| u × v \| = 2 Therefore, = +/- = +/-[0, 0,-1] We can now finaly calculate MN using; MN = AB . = [0, 2, 2] . +/-[0, 0, -1] = +/-( 0 + 0 + -2) = +/- 2 Since MN is a length we must take the positive result; The required distance is therefore 2 units

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The Perpendicular Distance between two Skew Lines

Problem:Find the perpendicular distance between the line passing through the the point (1, -1, 1) which is parallel to the vector u =[1, 3, 0] and the line passing through the point (1, 1, 3) which is parallel to the vector v = [1, 1, 0].

Problem:

Find the perpendicular distance between the line passing through the the point (1, -1, 1) which is parallel to the vector u =[1, 3, 0] and the line passing through the point (1, 1, 3) which is parallel to the vector v = [1, 1, 0].