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Three Dimensional Geometry

Question
CBSEENMA12033345

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

Solution
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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