Three Dimensional Geometry

Question

Find the vector and cartesian form of the equation of the plane containing two lines:
$straight r with rightwards arrow on top space equals straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis$
and $straight r with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top space plus space straight mu space left parenthesis negative 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 8 space straight k with hat on top right parenthesis$

Solution

Here,    $straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space minus 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 8 space straight k with hat on top$
The vector equation of plane is
$left parenthesis straight r with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis. space space open parentheses straight b with rightwards arrow on top space cross times straight c with rightwards arrow on top close parentheses space equals space 0$
or    $open square brackets straight r with rightwards arrow on top space minus space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis close square brackets. space open square brackets left parenthesis 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis space cross times space space left parenthesis negative 2 space straight i with hat on top space plus space space 3 space straight j with hat on top space plus space space 8 space straight k with hat on top close square brackets space equals space 0$
Cartesian equation of plane is
$open vertical bar table row cell straight x minus 1 end cell cell straight y minus 2 end cell cell straight z plus 4 end cell row 2 3 6 row cell negative 2 end cell 3 8 end table close vertical bar space equals space 0$
$or space space left parenthesis straight x minus 1 right parenthesis thin space open vertical bar table row 3 6 row 3 8 end table close vertical bar space minus space left parenthesis straight y minus 2 right parenthesis space open vertical bar table row 2 6 row cell negative 2 end cell 8 end table close vertical bar space plus space left parenthesis straight z plus 4 right parenthesis space open vertical bar table row 2 3 row cell negative 2 end cell 3 end table close vertical bar space equals space 0$
or    (x – 1) (24 – 18) – (y – 2) (16 + 12) + (z + 4) (6 + 6) = 0
or    6 (x – 1) – 28 (y – 2) + 12 (z + 4) = 0
or    3 (x – 1) – 14 (y – 2) + 6 (z + 4) = 0
or    3 x – 3 – l4y + 28 + 6z + 24 = 0
or    3 x – 14 y + 6 z + 49 = 0

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