How do you find the shortest distance between skew lines?

One way is to use vectors. The cross product of a vector along one line with a vector along the other line is a vector perpendicular to both lines. Divide that by its length to make a unit vector. Then project any vector that goes from one line to the other onto that perpendicular vector, by taking the dot product with the unit vector.

Another method is to change the name of the parameter to say s for one of the lines (so the parameters can vary independently), then use the distance formula to express the squared distance from an arbitrary point on the one line to an arbitrary point on the other line. That will be a quadratic function in s and t. Its partial derivatives must be zero where the distance is minimum, and they are a system of two linear equations in s and t. Solve them and plug into the squared distance expression. Then take the square root.

of route. there might want to nicely be an more advantageous elementary way, yet i'm interpreting optimization complications excellent now, so because it quite is how i'm in possibility of imagine of about this difficulty. initiate via writing an expression for the length of the line section from a element on one line to a element on different one. The positions of the criteria might want to need to be expressed in words of a unmarried variable. in case you used the slope/intercept formula for a line in 2 dimensions, y = mx + b, you may be in a position to need to certain a coordinate on a line as (x, mx + b) and verify that element replaced into on your line. It honestly might want to not be too difficult to certain an same element in additional advantageous-dimensional aspects. you in undemanding words might want to need to mission the line onto the planes of the self protecting variable you %. and one depending variable at a time. as instantly as you've were given your 2 aspects expressed as applications of a unmarried variable each and every, you go with to apply the Pythagorean theorem (or its more advantageous-dimensional analog) to locate the section between both aspects. you would possibly want to now write out this difficulty as a minimization. allow's say you said because the strains Line a million and Line 2 (inventive, i recognize) and also you named their coordinates with the line form as a subscript. you may be in a position to need to declare that you attempt to shrink over x1 and x2, the function for the section between aspects. Take the gradient of the function. The values of x1 and x2 that make it 0 provide you the places on the strains.