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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    Answer
    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.
                   

          
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    Answer
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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
                              

          

    The equations of two lines are
                    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                           ...(1)
    and           straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space minus 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                       ...(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 stack straight b subscript 1 with rightwards arrow on top space 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 comma space we space get comma
    stack straight a subscript 1 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 comma space space space space space stack straight b subscript 1 with rightwards arrow on top space equals space straight i 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 stack straight a 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 minus space 2 space straight k with hat on top comma space space stack straight b subscript 2 with rightwards arrow on top space equals space minus 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 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 2 space straight i with hat on top space plus space space 11 space straight j with hat on top
    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 1 3 2 row cell negative 1 end cell 2 1 end table close vertical bar space equals space left parenthesis negative 3 minus 4 right parenthesis space straight i with hat on top space minus space left parenthesis 1 plus 2 right parenthesis space straight j with hat on top space plus space left parenthesis 2 plus 3 right parenthesis space straight k with hat on top
                    equals space minus straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top
    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 1 plus 9 plus 25 end root space equals space square root of 35
    S.D.  = open vertical bar fraction numerator left parenthesis stack straight a subscript 2 with rightwards arrow on top minus stack straight a subscript 1 with rightwards arrow on top right parenthesis. space open parentheses stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close parentheses 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 close vertical bar space equals space open vertical bar fraction numerator left parenthesis 2 space straight i with hat on top space plus space 11 space straight j with hat on top right parenthesis. space left parenthesis negative straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis over denominator square root of 35 end fraction close vertical bar
            equals space open vertical bar fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis negative 1 right parenthesis thin space plus space left parenthesis 11 right parenthesis thin space left parenthesis negative 3 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 35 end fraction close vertical bar space equals space open vertical bar negative fraction numerator 35 over denominator square root of 35 end fraction close vertical bar space equals space square root of 35
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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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    Answer
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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
    Answer
    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
    73 Views
    Answer
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