Three Dimensional Geometry

Question

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector $2 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 3 space straight k with hat on top$ .

Solution

Let $straight n with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 3 space straight k with hat on top$
$therefore space space space space space straight n with hat on top space equals space fraction numerator straight n with rightwards arrow on top over denominator open vertical bar straight n with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator 2 straight i with hat on top space plus space 6 straight j with hat on top space minus space 3 space straight k with hat on top over denominator square root of 4 plus 36 plus 9 end root end fraction space equals space 1 over 7 open parentheses 2 straight i with hat on top space plus space 6 straight j with hat on top space minus space 3 straight k with hat on top close parentheses$
$equals space 2 over 7 straight i with hat on top space plus space 6 over 7 straight j with hat on top space minus space 3 over 7 straight k with hat on top$
∴ required equation of plane is
$straight r with rightwards arrow on top. space space open parentheses 2 over 7 straight i with hat on top space plus space 6 over 7 straight j with hat on top space minus space 3 over 7 straight k with hat on top close parentheses space equals 5$
or $straight r with rightwards arrow on top space. space open parentheses 2 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 3 space straight k with hat on top close parentheses space equals space 35$

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