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Three Dimensional Geometry

Long Answer Type

281. Find the vector and cartesian equation of the plane containing the two lines
$straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis$
and $straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space space 3 space straight j with hat on top space minus space 7 space straight k with hat on top space plus space straight mu space left parenthesis 3 straight i with hat on top space minus space 2 space straight j with hat on top space plus space space 5 space straight k with hat on top right parenthesis$

$straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis$
and $straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space space 3 space straight j with hat on top space minus space 7 space straight k with hat on top space plus space straight mu space left parenthesis 3 straight i with hat on top space minus space 2 space straight j with hat on top space plus space space 5 space straight k with hat on top right parenthesis$
$therefore space space space space straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top comma space space space space straight a with rightwards arrow on top apostrophe space equals space 3 space straight i with hat on top space plus space space 3 space straight j with hat on top space minus space 7 space straight k with hat on top comma$
   $straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 5 space straight k with hat on top comma space space space space straight c with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top$
The vector equation of the plane is
$open parentheses straight r with rightwards arrow on top space minus space straight a with rightwards arrow on top close parentheses. space space open parentheses straight b with rightwards arrow on top cross times space straight c with rightwards arrow on top close parentheses space equals space 0$
or    $open square brackets straight r with rightwards arrow on top space minus space left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis close square brackets. space space space open square brackets left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 5 space straight k with hat on top right parenthesis space cross times space left parenthesis 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis close square brackets space equals space 0$
The cartesian equation of plane is
$open vertical bar table row cell straight x minus 2 end cell cell straight y minus 1 end cell cell straight z plus 3 end cell row 1 2 5 row 3 cell negative 2 end cell 5 end table close vertical bar space equals space 0$
or    $left parenthesis straight x minus 2 right parenthesis thin space open vertical bar table row 2 5 row cell negative 2 end cell 5 end table close vertical bar space space minus space left parenthesis straight y minus 1 right parenthesis space open vertical bar table row 1 5 row 3 5 end table close vertical bar space plus space left parenthesis straight z plus 3 right parenthesis space open vertical bar table row 1 2 row 3 cell negative 2 end cell end table close vertical bar space equals space 0$
or    (x – 2) (10 + 10) – (y – 1) (5 – 15) + (z + 3) (–2 – 6) = 0
or    20 (x – 2) + 10 (y – 1) – 8 (z + 3) = 0
or    10 (x – 2) + 5 (y – 1) – 4 (z + 3) = 0
or    10x – 20 + 5y – 5 – 4z – 12 = 0
or    10x + 5y – 4 z = 37