The equation of the plane is given by the vector equation $\left(\mathbf{R}-\mathbf{P}\right)·\mathbf{N}\=0$, that is, by

$\(\left[\begin{array}{c}x\\ y\\ z\end{array}\right]-\left[\begin{array}{c}5\\ -3\\ 7\end{array}\right]\)·\left[\begin{array}{c}3\\ -5\\ 2\end{array}\right]$ = $3\left(x-5\right)-5\left(y\+3\right)\+2\left(z-7\right)\=0$

which simplifies to $3 x-5 y\+2 z\=15\+15\+14\=44$.

Alternatively, some practitioners would write $\mathbf{N}·\mathbf{R}\=d$ and use point P to determine the value of $d$. The calculation we go as follows.

$\genfrac{}{}{0ex}{}{\left(3 x-5 y\+2 z\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=5\,y\=-3\,z\=7}\=d$ ⇒ $d\=15\+15\+14\=44$

Hence, the equation of the plane is $3 x-5 y\+2 z\=44$.

The traditional vector approach is implemented below.