Three Dimensional Geometry

Find the shortest distance between the lines: - Wired Faculty

Question

Find the shortest distance between the lines:
$fraction numerator straight x plus 1 over denominator 7 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 6 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction space and space fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 7 over denominator 1 end fraction$

Solution

The equations of two lines are

fraction numerator straight x plus 1 over denominator 7 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 6 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction space and space fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 7 over denominator 1 end fraction

fraction numerator straight x minus left parenthesis negative 1 right parenthesis over denominator 7 end fraction space equals space fraction numerator straight y minus left parenthesis negative 1 right parenthesis over denominator negative 6 end fraction space equals space fraction numerator straight z minus left parenthesis negative 1 right parenthesis over denominator 1 end fraction space and space fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 7 over denominator 1 end fraction

therefore

straight x subscript 1 space equals space minus 1 comma space space space straight y subscript 1 space equals space minus 1 comma space space straight z subscript 1 space equals space minus 1 semicolon space space straight a subscript 1 space equals space 7 comma space space straight b subscript 1 space equals space minus 6 comma space space straight c subscript 1 space equals space 1

and

straight x subscript 2 space equals space 3 comma space space space straight y subscript 2 equals 5 comma space space space straight z subscript 2 space equals space 7 semicolon space space space straight a subscript 2 space equals space 1 comma space space straight b subscript 2 space equals space minus 2 comma space space straight c subscript 2 space equals 1

therefore space space space space straight x subscript 2 minus straight x subscript 1 space equals space 4 comma space space space space space straight y subscript 2 minus straight y subscript 1 space equals space 6 comma space space space straight z subscript 2 space minus space straight z subscript 1 space equals space 8

therefore space space space space open vertical bar table row cell straight x subscript 2 minus straight x subscript 1 end cell cell straight y subscript 2 minus straight y subscript 1 end cell cell straight z subscript 2 minus straight z subscript 1 end cell row cell straight a subscript 1 end cell cell straight b subscript 1 end cell cell straight c subscript 1 end cell row cell straight a subscript 2 end cell cell straight b subscript 2 end cell cell straight c subscript 2 end cell end table close vertical bar space equals space open vertical bar table row 4 6 8 row 7 cell negative 6 end cell 1 row 1 cell negative 2 end cell 1 end table close vertical bar

equals space 4 space open vertical bar table row cell negative 6 end cell 1 row cell negative 2 end cell 1 end table close vertical bar space minus space 6 space open vertical bar table row 7 1 row 1 1 end table close vertical bar space plus space 8 space open vertical bar table row 7 cell negative 6 end cell row 1 cell negative 2 end cell end table close vertical bar

equals space 4 space left parenthesis negative 6 plus 2 right parenthesis space minus space 6 space left parenthesis 7 minus 1 right parenthesis space plus space 8 thin space left parenthesis negative 14 plus 6 right parenthesis equals space 4 left parenthesis negative 4 right parenthesis space minus space 6 space left parenthesis 6 right parenthesis space plus space 8 space left parenthesis negative 8 right parenthesis equals space minus 16 minus 36 minus 64 space equals space minus 116

square root of left parenthesis straight b subscript 1 straight c subscript 2 minus straight b subscript 2 straight c subscript 1 right parenthesis squared plus left parenthesis straight c subscript 1 straight a subscript 2 minus straight c subscript 2 straight a subscript 1 right parenthesis squared plus left parenthesis straight a subscript 1 straight b subscript 2 minus straight a subscript 2 straight b subscript 1 right parenthesis squared end root

equals space square root of left parenthesis negative 6 plus 2 right parenthesis squared plus left parenthesis 1 minus 7 right parenthesis squared plus left parenthesis negative 14 plus 6 right parenthesis squared end root space equals space square root of 16 plus 36 plus 64 end root space equals space square root of 116

therefore space space space straight S. straight D. space equals space open vertical bar fraction numerator negative 116 over denominator square root of 116 end fraction close vertical bar space equals fraction numerator 116 over denominator square root of 116 end fraction space equals square root of 116 space equals space square root of 4 cross times 29 end root space equals space 2 square root of 29.