Shortest Distance Between Two Skew Lines

Evaluate |AB X CD| where A is (6, -3, 0), B is (3, -7, 1), C is (3, 7, -1) and D is (4,5,-3). Hence find the shortest distance between AB and CD

The two key points two remember here are:

  1. The shortest line between two lines is perpendicular to both
  2. When two vectors are crossed, the result is a vector that is perpendicular to both

Thus the vector representing the shortest distance between AB and CD will be in the same direction as (AB X CD), which can be written as a constant times (AB X CD). We can then equate this vector to a general vector between two points on AB and CD, which can be obtained from the vector equations of the two lines. Solving the three equations obtained simultaneously, we can find the constant and the shortest distance.

Here is my solution:


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