given 2 lines with equations:

the shortest distance between them can be found by 3 methods.

***** method 1 ******

***** method 1 ******

the shortest distance line make right angle with 2 given lines. Thus, bu simple geometry, we have

where is the distance between 2 lines. there are 2 unknowns : . the 3 components of the vectors give 3 equations. Thus, by solving the matrix equation:

Notice that the matrix N is formed by 3 independence vectors when the 2 given lines is not parallel, therefore, it has an inverse and the there is always an unique solution.

***** method 2 ******

another method is minimized the distance between 2 lines. the distance between 2 lines is:

by minimize the distance square for variate the parameter , i.e.

by solving the matrix equation:

solving and get . then get .

*********** remark *****************

it is interesting that, the method 2 separate the problem into 2 parts, while method 1 solve it at once. And reduced a 3 dimensional problem into 2 dimensional.

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