LinAlg4.2.7

Find the equation of the plane passing through the point $(3,4,5)\,$ and parallel to the plane $\bar{r}\cdot(2\bar{i}+3\bar{j}-\bar{k})=6\,$

Given plane is $\bar{r}\cdot(2\bar{i}+3\bar{j}-\bar{k})=6\,$

Let $\pi\,$ be the required plane.Given this is parallel to the above plane.

But $\pi\,$ passes through $(3,4,5)\,$ and is perpendicular to $2\bar{i}+3\bar{j}-\bar{k}\,$

Therefore vector equation is $(\bar{r}-\bar{a})\cdot\bar{n}=0\,$

$[\bar{r}-(3\bar{i}+4\bar{j}+5\bar{k})]\cdot(2\bar{i}+3\bar{j}-\bar{k})=0\,$

Simplifying we get

$\bar{r}\cdot(2\bar{i}+3\bar{j}-\bar{k})=6+12-5=13\,$