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

    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.
          
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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.
                   

          

    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 space space and space 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 comma 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 space space 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 stack straight b subscript 1 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 the point on the line (1) with position vector stack straight a subscript 1 with rightwards arrow on top and T be the point on the line (2) with position vector stack straight a subscript 2 with rightwards arrow on top comma space so space that
    ST with rightwards arrow on top space equals space 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 space cross times space 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 space equals 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
                     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

    therefore          open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times space 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 space square root of 6

                                    equals space minus space 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 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 space cross times space stack straight b subscript 2 with rightwards arrow on top.
    If straight n with rightwards arrow on top is a vector along PQ with rightwards arrow on top comma then
                   straight n with rightwards arrow on top space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top space cross times space 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 space 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 = Projection of 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 space equals space open parentheses 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 close parentheses. 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
                      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
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    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
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    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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