Find the distance between the lines l1, and l2 given by     

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 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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 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.
               

      

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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
               
      

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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.


 
               
      

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 Multiple Choice QuestionsShort Answer Type

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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


The equations of 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 minus space 4 space straight k with hat on top space plus space straight lambda space left parenthesis 2 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             ...(1)
and                   straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus 3 space 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         ...(2)
Line (1) passes through the point S with position vector 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 minus space 4 space straight k with hat on top and line (2) passes through the point T with position vector stack straight a subscript 2 with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top. Also both lines are parallel to the vector straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus 3 space straight j with hat on top space plus 6 space straight k with hat on top.
Now,          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 straight j with hat on top space minus space straight k with hat on top
straight b with rightwards arrow on top cross times space left parenthesis stack straight a subscript 2 with rightwards arrow on top space minus space stack straight a subscript 1 with rightwards arrow on top right parenthesis 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 6 row 2 1 cell negative 1 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 3 6 row 1 cell negative 1 end cell end table close vertical bar space minus space straight j with hat on top open vertical bar table row 2 6 row 2 cell negative 1 end cell end table close vertical bar space plus space straight k with hat on top open vertical bar table row 2 3 row 2 1 end table close vertical bar
                          equals space left parenthesis negative 3 minus 6 right parenthesis space straight i with hat on top space minus space left parenthesis negative 2 space minus space 12 right parenthesis space straight j with hat on top space plus space left parenthesis 2 minus 6 right parenthesis space straight k with hat on top
equals space minus 9 straight i with hat on top space plus space 14 space straight j with hat on top space minus space 4 space straight k with hat on top

           open vertical bar straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 2 right parenthesis squared plus left parenthesis 3 right parenthesis squared plus left parenthesis 6 right parenthesis squared end root space equals space square root of 4 plus 9 plus 36 end root space equals space square root of 49 space equals space 7
Let d be the distance between the lines
therefore space space space space space space straight d space equals space open vertical bar fraction numerator straight b with rightwards arrow on top cross times 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 over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction close vertical bar space equals space fraction numerator open vertical bar negative 9 space straight i with hat on top space plus space 1 space 4 space straight j with hat on top space minus space 4 space straight k with hat on top close vertical bar over denominator 7 end fraction

             equals space fraction numerator square root of left parenthesis negative 9 right parenthesis squared plus left parenthesis 14 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root over denominator 7 end fraction space equals space fraction numerator square root of 81 plus 196 plus 16 end root over denominator 7 end fraction
equals space fraction numerator square root of 293 over denominator 7 end fraction space equals space 1 over 7 square root of 293
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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

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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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