Three Dimensional Geometry

Find the cartesian equation of the plane through the point with position vector and perpendicular to the vector . - Wired Faculty

Question

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Find the cartesian equation of the plane through the point with position vector $2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top$ and perpendicular to the vector $4 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top$ .

Solution

Here, $stack straight r subscript 1 with rightwards arrow on top space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top comma space space space space straight n with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top$
$therefore space space space space space space space the space equation space of space plane space is space straight n with rightwards arrow on top. space left parenthesis straight r with rightwards arrow on top space minus space stack straight r subscript 1 with rightwards arrow on top right parenthesis space equals space 0$
$or space space left parenthesis 4 straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis space. open curly brackets left parenthesis straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top right parenthesis space minus space left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top right parenthesis close curly brackets space equals space 0$
$or space left parenthesis 4 straight i with hat on top space plus space 2 straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis. space space space open curly brackets left parenthesis straight x minus 2 right parenthesis space straight i with hat on top space plus space left parenthesis straight y plus 1 right parenthesis space straight j with hat on top space plus space left parenthesis straight z minus 1 right parenthesis space straight k with hat on top close curly brackets space equals space 0 or space 4 space left parenthesis straight x minus 2 right parenthesis space plus space 2 left parenthesis straight y plus 1 right parenthesis space minus space 3 space left parenthesis straight z minus 1 right parenthesis space equals space 0 or space space 4 straight x minus 8 plus 2 straight y plus 2 minus 3 straight z plus 3 space equals space 0 or space space 4 straight x plus 2 straight y minus 3 straight z space equals space 3$
This is cartesian equation of plane.

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Three Dimensional Geometry

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