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Three Dimensional Geometry

Question
CBSEENMA12033496

Show that the lines
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 minus space straight k with hat on top space plus space straight lambda open parentheses 3 space straight i with hat on top space minus space straight j with hat on top close parentheses space space and space 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 open parentheses 2 straight i with hat on top space plus space 3 space straight k with hat on top close parentheses intersect.

Solution

The given lines 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 minus space straight k with hat on top space plus space straight lambda open parentheses 3 straight i with hat on top space minus space straight j with hat on top close parentheses                                 ...(1)

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 left parenthesis 2 straight i with hat on top space plus space 3 space straight k with hat on top right parenthesis                                ...(2)
therefore space space space space straight a with 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 comma space space straight b with rightwards arrow on top space equals space 3 straight i with hat on top space minus space straight j with hat on top comma space space straight c 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 comma space space straight d with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight k with hat on top
therefore space space space space open square brackets straight c with rightwards arrow on top space minus space stack straight a comma with rightwards arrow on top space straight b with rightwards arrow on top comma space straight d with rightwards arrow on top close square brackets space equals space open vertical bar table row 3 cell negative 1 end cell 0 row 3 cell negative 1 end cell 0 row 2 0 3 end table close vertical bar space equals space 3 space open vertical bar table row 3 cell negative 1 end cell row 3 cell negative 1 end cell end table close vertical bar space equals space 3 left parenthesis negative 3 plus 3 right parenthesis space equals space 3 left parenthesis 0 right parenthesis space equals space 0
∴ lines (1) and (2) are coplanar
But
straight b 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 space and space straight d with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight k with hat on top are not parallel
∴  lines (1) and (2) intersect.