Find the vector equation of a plane, which is at a distance of 5 units from the origin and whose normal vector is $2\hat{i}-\hat{j}+2\hat{k}\cdot$

$\vec{n}= 2\hat{i}-\hat{j}+2\hat{k}$
we know that  $\vec{n}=\frac{\vec{n}}{\left | \vec{n} \right |}= \frac{ 2\hat{i}-\hat{j}+2\hat{k}}{\sqrt{\left ( 2 \right )^{2}+\left ( -1 \right )^{2}+\left ( 2 \right )^{2}}}$
$= \frac{ 2\hat{i}-\hat{j}+2\hat{k}}{3}= \frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}$
$r\cdot \vec{n}= p$  where p is the distance from origin
$\Rightarrow r\cdot \left [\frac{2}{3}\hat{i} -\frac{1}{3}\hat{j} +\frac{2}{3}\hat{k}\right ]= 5$
$\Rightarrow r \left [2\hat{i}-\hat{j}+2\hat{k}]= 15$

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