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Equation of the line of the shortest distance between the lines x/2=y/-3=z/1 and x-2/3=y-1/-5=z+2/2 is

Asked by aryamansingh1123 15th October 2017, 10:38 AM
Answered by Expert
Answer:
begin mathsize 16px style Consider space first space equation space of space the space line
straight x over 2 equals fraction numerator straight y over denominator negative 3 end fraction equals straight z over 1
rightwards double arrow fraction numerator straight x minus 0 over denominator 2 end fraction equals fraction numerator straight y minus 0 over denominator negative 3 end fraction equals fraction numerator straight z minus 0 over denominator 1 end fraction
Comparing space with space
fraction numerator straight x minus straight x subscript 1 over denominator straight a subscript 1 end fraction equals fraction numerator straight y minus straight y subscript 1 over denominator straight b subscript 1 end fraction equals fraction numerator straight z minus straight z subscript 1 over denominator straight c subscript 1 end fraction space we space get
rightwards double arrow straight x subscript 1 space equals space 0 comma space straight y subscript 1 space equals space 0 comma space straight z subscript 1 space equals space 0 comma space straight a subscript 1 space equals space 2 comma space straight b subscript 1 space equals space minus 3 comma space straight c subscript 1 space equals space 1
Consider space second space equation space of space the space line
fraction numerator straight x minus 2 over denominator 3 end fraction equals fraction numerator straight y minus 1 over denominator negative 5 end fraction equals fraction numerator straight z plus 2 over denominator 2 end fraction
rightwards double arrow fraction numerator straight x minus 2 over denominator 3 end fraction equals fraction numerator straight y minus 1 over denominator negative 5 end fraction equals fraction numerator straight z minus open parentheses negative 2 close parentheses over denominator 2 end fraction
Comparing space with space
space fraction numerator straight x minus straight x subscript 2 over denominator straight a subscript 2 end fraction equals fraction numerator straight y minus straight y subscript 2 over denominator straight b subscript 2 end fraction equals fraction numerator straight z minus straight z subscript 2 over denominator straight c subscript 2 end fraction space
rightwards double arrow straight x subscript 2 space equals space 2 comma space straight y subscript 2 space equals space 1 comma space straight z subscript 2 space equals space minus 2 comma space straight a subscript 2 space equals space 3 comma space straight b subscript 2 space equals space minus 5 comma space straight c subscript 2 space equals space 2
Using space shortest space distance space formula comma
open vertical bar fraction numerator open vertical bar table row cell straight x subscript 2 space minus straight x subscript 1 end cell cell straight y subscript 2 space minus space 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 over denominator square root of open parentheses straight b subscript 1 straight c subscript 2 minus straight b subscript 2 straight c subscript 1 close parentheses squared plus open parentheses straight c subscript 1 straight a subscript 2 minus straight a subscript 1 straight c subscript 2 close parentheses squared plus open parentheses straight a subscript 1 straight b subscript 2 minus straight a subscript 2 straight b subscript 1 close parentheses squared end root end fraction close vertical bar
You space can space solve space it space further.

NOTE colon space
open vertical bar table row cell straight y subscript 2 space minus space straight y subscript 1 end cell cell straight z subscript 2 minus straight z subscript 1 end cell row cell straight b subscript 1 end cell cell straight c subscript 1 end cell row cell straight b subscript 2 end cell cell straight c subscript 2 end cell end table close vertical bar equals space open parentheses straight b subscript 1 straight c subscript 2 minus straight b subscript 2 straight c subscript 1 close parentheses squared space rightwards arrow first space term space in space the space square space root
Similarly space you space can space do space it space for space the space remaining space terms space to space remember space the space formula.
If space you space change space the space order space then space also space value space of space the space denominator space will space not space change space as
it space is space straight a space square space and space square space is space always space positive.
end style
Answered by Expert 19th December 2017, 5:32 PM
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