Find the shortest distance between the lines:            �
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    Three Dimensional Geometry

    Multiple Choice QuestionsShort Answer Type

    121. Find the shortest distance between the lines:
          straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 15 space straight j with hat on top space plus space 9 space straight k with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis
    and straight r with rightwards arrow on top space equals space left parenthesis 2 space straight mu minus 1 right parenthesis space straight i with hat on top space plus space left parenthesis 1 plus straight mu right parenthesis space straight j with hat on top space plus space left parenthesis 9 minus 3 space straight mu right parenthesis space straight k with hat on top
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    Answer

    Multiple Choice QuestionsLong Answer Type

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    122. Find the shortest distance between the lines:
                   straight r with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top space plus space straight lambda space open parentheses 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses
    and         straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses.
          

    The equations of the two lines are
                      straight r with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top space plus space straight lambda space open parentheses 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses                       ...(1)
    and            straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses                    ...(2)
    Comparing these equations with
           straight r with rightwards arrow on top space equals space stack straight a subscript 1 with rightwards arrow on top space plus space straight lambda space stack straight b subscript 1 with rightwards arrow on top space and space straight r with rightwards arrow on top space equals space stack straight a subscript 2 with rightwards arrow on top space plus space straight mu space stack straight b subscript 2 with rightwards arrow on top
    We get,
            stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top comma space space stack straight b subscript 1 with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top stack straight a subscript 2 with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top comma space space stack straight b subscript 2 with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top
    Let S be point on line (1) with position vector stack straight a subscript 1 with rightwards arrow on top and T be point on line (2) with position vector stack straight a subscript 2 with rightwards arrow on top so that
       
                     ST with rightwards arrow on top space equals stack straight a subscript 2 with rightwards arrow on top space minus space stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top

    Now, stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 2 3 4 row 3 4 5 end table close vertical bar
                          equals space left parenthesis 15 minus 16 right parenthesis space straight i with hat on top space minus space left parenthesis 10 minus 12 right parenthesis space straight j with hat on top space plus space left parenthesis 8 minus 9 right parenthesis space straight k with hat on top space equals negative straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top

     therefore space space space open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close vertical bar space equals space square root of left parenthesis negative 1 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared end root space equals square root of 6

    Let PQ with rightwards arrow on top be the S.D. vector between given lines. Therefore, it is parallel to stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top.
    If straight n with rightwards arrow on top is a unit vector along PQ with rightwards arrow on top, 
    then
                        straight n with rightwards arrow on top space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close vertical bar end fraction space equals fraction numerator 1 over denominator square root of 6 end fraction left parenthesis negative straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top right parenthesis
    Now S.D. = Projection of ST with rightwards arrow on top space on space PQ with rightwards arrow on top space equals space Projection space of space ST with rightwards arrow on top space on space straight n with rightwards arrow on top space equals space ST with rightwards arrow on top. space straight n with rightwards arrow on top
                     equals space open parentheses straight i with hat on top space plus 2 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses space. space fraction numerator 1 over denominator square root of 6 end fraction space open parentheses negative straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top close parentheses space equals space fraction numerator 1 over denominator square root of 6 end fraction left parenthesis negative 1 plus 4 minus 2 right parenthesis space equals space fraction numerator 1 over denominator square root of 6 end fraction

                         equals space left parenthesis 15 minus 16 right parenthesis space straight i with hat on top space space minus space left parenthesis 10 minus 12 right parenthesis space straight j with hat on top space plus space left parenthesis 8 minus 9 right parenthesis space straight k with hat on top space equals space minus straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top
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    123. Find the shortest distance between the lines:
                   straight r with rightwards arrow space on top equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top space plus space straight lambda space open parentheses 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses
    and       straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses.
                   

          
    73 Views
    Answer
    124. Find the shortest distance between the lines:
                   straight r with rightwards arrow on top space equals space straight i with hat on top space minus space 7 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight lambda space open parentheses stack straight i space with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses
    and         straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight mu space open parentheses negative straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses
                              

          
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    125. Find the shortest distance between the lines:
    straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis
    and straight r with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top space minus space straight mu open parentheses 2 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top close parentheses
                   
          
    77 Views
    Answer
    126. Find the shortest distance between the lines:
             straight r with rightwards arrow on top space equals space left parenthesis 8 plus 3 space straight lambda right parenthesis space straight i with hat on top space minus space left parenthesis 9 plus 16 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 10 plus space 7 space straight lambda right parenthesis space straight k with hat on top
    and   straight r with rightwards arrow on top space equals space 15 space straight i with hat on top space plus space 29 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 8 space straight j with hat on top space minus space 5 space straight k with hat on top close parentheses.


     
                   
          
    77 Views
    Answer

    Multiple Choice QuestionsShort Answer Type

    127. Find the distance between the lines l1, and l2 given by
                          straight r with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space space 6 space straight k with hat on top right parenthesis
    and                straight r with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 3 straight j with hat on top space minus space 5 space straight k with hat on top space plus space straight mu space left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis
    84 Views
    Answer

    Multiple Choice QuestionsLong Answer Type

    128. 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
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    129. Determine whether the following pair of lines intersect:
          straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top right parenthesis
    and straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight mu left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis
    72 Views
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    130. Determine whether the following pair of lines intersect:
               straight r with space rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top space plus space straight lambda left parenthesis 3 space straight i with hat on top space minus space straight j with hat on top right parenthesis
    and       straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space straight k with hat on top space plus space straight mu space left parenthesis 2 straight i with hat on top space plus space 3 space stack straight k right parenthesis with hat on top
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    Answer
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