Find the shortest distance between the lines:      and . f
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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
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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
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    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.

    The equations of the two lines are
              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                        ...(1)
    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         ...(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 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 3 space straight i with hat on top space plus space 8 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 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
    stack straight a subscript 2 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 space 6 space straight k with hat on top comma space space stack straight b subscript 2 with rightwards arrow on top space equals space minus 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
    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 minus 6 space straight i with hat on top space minus space 15 space straight j with hat on top space plus space 3 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 3 cell negative 1 end cell 1 row cell negative 3 end cell 2 4 end table close vertical bar
                                equals space left parenthesis negative 4 minus 2 right parenthesis space straight i with hat on top space minus space left parenthesis 12 plus 3 right parenthesis space straight j with hat on top space plus space left parenthesis 6 minus 3 right parenthesis space straight k with hat on top space equals space minus 6 space straight i with hat on top space minus space 15 space straight j with hat on top space plus space 3 space straight k with hat on top
    therefore      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 36 plus 225 plus 9 end root space equals space square root of 270 space equals space 3 square root of 30
    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 3 square root of 30 end fraction left parenthesis negative 6 space straight i with hat on top space minus space 15 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 30 end fraction left parenthesis negative space 2 space straight i with hat on top space minus space 5 space straight j with hat on top space plus 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 stack ST. with rightwards arrow on top straight n with rightwards arrow on top
                     equals space open parentheses negative 6 space straight i with hat on top space minus space 15 space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses space. space fraction numerator 1 over denominator square root of 30 end fraction left parenthesis negative 2 straight i with hat on top space minus space 5 space straight j with hat on top space plus space straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 30 end fraction left parenthesis 12 plus 75 plus 3 right parenthesis space equals space fraction numerator 90 over denominator square root of 30 end fraction space equals space fraction numerator 3 cross times 30 over denominator square root of 30 end fraction space equals space 3 square root of 30 space units
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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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