Three Dimensional Geometry

Question

Find the vector and cartesian equations of the planes:
that passes through the point (1, 4, 6) and the normal vector to the plane is $straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top$ .

Solution

Here, $stack straight r subscript 1 with rightwards arrow on top space equals space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top comma space space space straight n with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top$
The vector equation of plane is
$open parentheses straight r with rightwards arrow on top space minus space stack straight r subscript 1 with rightwards arrow on top close parentheses. space space straight n with rightwards arrow on top space equals space 0$
or $open square brackets straight r with rightwards arrow on top space minus space left parenthesis straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis close square brackets space. space open parentheses straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses space equals space 0$
Take $straight r space equals space straight x space straight i with hat on top space space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top$
$therefore space space space space open square brackets left parenthesis straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis close square brackets. space left parenthesis straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top right parenthesis space equals space 0$
or $open square brackets left parenthesis straight x minus 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight y minus 4 right parenthesis space straight j with hat on top space plus space left parenthesis straight z minus 6 right parenthesis space straight k with hat on top close square brackets. space space left parenthesis straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top right parenthesis space equals space 0$
or $left parenthesis straight x minus 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis straight y minus 4 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus space left parenthesis straight z minus 6 right parenthesis thin space left parenthesis 1 right parenthesis space equals space 0$
or $straight x minus 1 minus 2 straight y plus 8 plus straight z minus 6 space equals space 0$
or $straight x minus 2 straight y plus straight z plus 1 space equals space 0$
which is cartesian equation of plane.

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