Three Dimensional Geometry

Question

Find the vector and cartesian equations of the planes:
that passes through the point (1, 0, – 2) and the normal to the plane is $straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top$ .

Solution

Here, $stack straight r subscript 1 with rightwards arrow on top space equals straight i with hat on top space minus space 2 space straight j with hat on top comma space space space space space space straight n with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top$
The vector equation of line is
   $left parenthesis stack straight r space with rightwards arrow on top space minus space stack straight r subscript 1 with rightwards arrow on top right parenthesis. space straight n with rightwards arrow on top space equals space 0$
or    $open square brackets straight r with rightwards arrow on top space minus space open parentheses straight i with hat on top space minus space 2 space straight k with hat on top close parentheses close square brackets. space left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space 0$
Take $straight r equals straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top$
$therefore space space space space open square brackets left parenthesis straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space minus space 2 space straight k with hat on top right parenthesis close square brackets. space space left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space 0$
or    $open square brackets open parentheses straight x minus 1 close parentheses space straight i with hat on top space plus space straight y space straight j with hat on top space plus space left parenthesis straight z plus 2 right parenthesis space straight k with hat on top close square brackets. space space left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space 0$
or    $left parenthesis straight x minus 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis straight y right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis straight z plus 2 right parenthesis thin space left parenthesis negative 1 right parenthesis space equals space 0$
or    $straight x minus 1 plus straight y minus straight z minus 2 space equals space 0$
or $straight x plus straight y minus straight z space equals space 3$
which is cartesian equation of plane.

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