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Three Dimensional Geometry

Short Answer Type

141. If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.

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142. If the line drawn from (4, –1, 2) meets a plane at right angles at the point (–10, 5, 4), then find the equation of the plane.

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143. Find the vector and Cartesian equation of the plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratios 2, 3, - 4.

Here $stack straight r subscript 1 with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space straight n with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space 4 space straight k with hat on top$
The vector equation of plane is $open parentheses straight r with rightwards arrow on top space minus space stack straight r subscript 1 with rightwards arrow on top close parentheses. space straight n with rightwards arrow on top space equals space 0$
or    $open square brackets straight r with rightwards arrow on top space minus space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis close square brackets. space space space left parenthesis 2 straight i with hat on top space plus space 3 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis space equals space 0$
Take    $straight r space equals space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top$
$therefore space space space space space open square brackets left parenthesis straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis close square brackets space. space left parenthesis space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis space equals space 0$
or    $2 left parenthesis straight x minus 1 right parenthesis space plus space 3 space left parenthesis straight y minus 2 right parenthesis space minus space 4 space left parenthesis straight z minus 3 right parenthesis space equals space 0$
or $2 straight x plus 3 straight y minus 4 straight z space equals space minus 4$
which is Cartesian equation of plane.

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144. Find the vector and cartesian equations of the plane which passes through the point (5, 2, – 4) and perpendicular to the line with direction ratios 2, 3, –1.

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