Find the shortest distance between the lines whose vector equati
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    Three Dimensional Geometry

    Multiple Choice QuestionsLong Answer Type

    111. Show that the st. lines whose direction-cosines are given by the equations u I + v m + w n = 0,f m n + g n l + h l m = 0 are
    (i) perpendicular if straight f over straight u plus straight g over straight v plus straight h over straight w equals space 0
    (ii) parallel if square root of straight u space straight f space end root space plus space square root of straight v space straight g end root space plus space square root of straight w space straight h end root space space equals space 0
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    112.

    Find the shortest distance (S.D.) between the lines:
    stack straight r 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 space plus space straight lambda space left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space plus space stack straight k right parenthesis with hat on top stack straight r subscript 2 with rightwards arrow on top space space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space straight k with hat on top space plus space straight mu space left parenthesis 3 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 right parenthesis.

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    113. Find the shortest distance between the lines whose vector equations are
              straight r with rightwards arrow on top space equals left parenthesis 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 right parenthesis space plus space straight lambda left parenthesis straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis
    and    straight r with rightwards arrow on top space equals space 4 straight i with hat on top space plus 5 space straight j with hat on top space plus space 6 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 straight k with hat on top right parenthesis
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    114. Find the shortest distance between the lines
                                     straight r with rightwards arrow on top space equals space 6 space straight i with hat on top space plus space 2 space straight j with hat on top space space plus space 2 space straight k with hat on top space plus space straight lambda left parenthesis straight i with hat on top space minus space 2 straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis
     and                          straight r with rightwards arrow on top space equals space minus 4 space straight i with hat on top space minus space straight k with hat on top space plus space straight mu left parenthesis 3 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis
                  
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    115.

    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 straight j with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space straight k 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

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    116. Find the shortest distance between the lines whose vector equations are:
    straight r with rightwards arrow on top space equals space left parenthesis 1 minus straight t right parenthesis space straight i with hat on top space plus space left parenthesis straight t minus 2 right parenthesis space straight j with hat on top space plus space left parenthesis 3 minus 2 space straight t right parenthesis space straight k with hat on top
    and straight r with rightwards arrow on top space equals space left parenthesis straight s plus 1 right parenthesis space straight i with hat on top space plus space left parenthesis 2 straight s minus 1 right parenthesis straight j with hat on top space plus space left parenthesis 2 straight s plus 1 right parenthesis space straight k with hat on top

    The equations of the two lines are
              straight r with rightwards arrow on top space equals space left parenthesis 1 minus straight t right parenthesis space straight i with hat on top space plus space left parenthesis straight t minus 2 right parenthesis space straight j with hat on top space plus space left parenthesis 3 minus 2 straight t right parenthesis space straight k with hat on top
    and    straight r with rightwards arrow on top space equals space left parenthesis straight s plus 1 right parenthesis space straight i with hat on top space plus space left parenthesis 2 straight s minus 1 right parenthesis space straight j with hat on top space minus space left parenthesis 2 straight s plus 1 right parenthesis space straight k with hat on top
    or       straight r with rightwards arrow on top space equals space left parenthesis straight i with hat on top space minus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space plus space straight t space left parenthesis negative straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis                        ...(1)
    and     straight r with rightwards arrow on top space equals space left parenthesis straight i with hat on top minus straight j with hat on top space minus space straight k with hat on top right parenthesis space plus space straight s thin 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                                 ...(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 t space space stack straight b subscript 1 with rightwards arrow on top space space space and space 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 s space stack straight b subscript 2 with rightwards arrow on top stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space stack straight b subscript 1 with rightwards arrow on top space equals space minus space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top stack straight a subscript 2 with rightwards arrow on top space space equals straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top comma space space stack straight b subscript 2 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 2 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 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 j with hat on top space minus space 4 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 cell negative 1 end cell 1 cell negative 2 end cell row 1 2 cell negative 2 end cell end table close vertical bar
                       equals space left parenthesis negative 2 plus 4 right parenthesis space straight i with hat on top space minus space left parenthesis 2 plus 2 right parenthesis space straight j with hat on top space space plus space left parenthesis negative 2 minus 1 right parenthesis space straight k with hat on top space equals space 2 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 3 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 4 plus 16 plus 9 end root space equals square root of 29
    Let PQ with rightwards arrow on top  be the S.D. vector between the 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 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 space fraction numerator 1 over denominator square root of 29 end fraction left parenthesis 2 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 3 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 left parenthesis straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis. space fraction numerator 1 over denominator square root of 29 end fraction left parenthesis 2 space straight i with hat on top space minus space 4 space space straight j with hat on top space minus space space 3 space straight k with hat on top right parenthesis
                   equals space fraction numerator 1 over denominator square root of 29 end fraction left parenthesis 0 minus 4 plus 12 right parenthesis space equals space fraction numerator 8 over denominator square root of 29 end fraction space units. space
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    117. Find the shortest distance between the lines whose vector equations are
            straight r with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k 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 plus space straight j with hat on top space minus space straight k with hat on top space plus space straight mu space left parenthesis 3 straight i with hat on top space minus space 5 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis
    75 Views
    Answer
    118. Find the shortest distance between the following 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 straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space minus space straight j with hat on top space plus space straight k 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 minus 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 straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis  
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    119. 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 plus space space 8 space straight j with hat on top space plus space 3 space straight k with hat on top space plus space straight lambda space left parenthesis 3 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top right parenthesis
    and straight r with rightwards arrow on top space equals space minus 3 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space 6 space straight k with hat on top space plus space straight mu space open parentheses negative 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses space.
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    120. Find the shortest distance between the lines
      straight r with rightwards arrow on top space equals space left parenthesis straight lambda minus 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight lambda plus 1 right parenthesis space straight j with hat on top space minus space left parenthesis straight lambda plus 1 right parenthesis space straight k with hat on top
    and straight r with rightwards arrow on top space equals space left parenthesis 1 minus straight mu right parenthesis space straight i with hat on top space plus space left parenthesis 2 straight mu minus 1 right parenthesis space straight j with hat on top space plus space left parenthesis straight mu space plus space 2 right parenthesis space straight k with hat on top
                                         
          
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