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Three Dimensional Geometry

Question
CBSEENMA12033495

Determine whether 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 open parentheses 2 straight i with hat on top space plus space straight k with hat on top close parentheses
and   straight r with rightwards arrow on top space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight mu space open parentheses straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses space space intersect.

Solution
The given lines are
                        straight r with rightwards arrow on top space equals straight i with hat on top space minus space straight j with hat on top space plus space straight lambda open parentheses 2 straight i with hat on top space plus space straight k with hat on top close parentheses                             ...(1)
and                  straight r with rightwards arrow on top space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight mu open parentheses straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses                    ...(2)
therefore space space space space space space space space space space space straight a with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight k with hat on top
                straight c 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 comma space space straight d 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
open square brackets straight c with rightwards arrow on top minus straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space space straight d with rightwards arrow on top close square brackets space equals space open vertical bar table row 1 0 0 row 2 0 1 row 1 1 cell negative 1 end cell end table close vertical bar space equals space 1 space open vertical bar table row 0 1 row 1 cell negative 1 end cell end table close vertical bar
                      equals space 1 space left parenthesis 0 minus 1 right parenthesis space equals space minus space 1 space not equal to 0
∴  lines {1) and (2) are not coplanar
∴   lines (1) and (2) do not intersect.

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