Find a cartesian equation for the plane containing the point (3, 1, 5) and perpendicular to the line with
parametric form x = 5t + 1, y = 6t − 2, z = 8t, t 2 R.
The direction vector of the line is the normal vector of the plane: $\displaystyle \overrightarrow{n_P}= (5,6,8)$
Let A(3,1,5) denote a point located in the plane with it's stationary vector $\displaystyle \vec a$ and $\displaystyle \vec p = (x,y,z)$ the stationary vector of any point in the plane then the equation of the plane P is:
$\displaystyle P: \overrightarrow{n_P} \cdot ( \vec p - \vec a) = 0$
Using the given values you have:
$\displaystyle P: (5,6,8)((x,y,z)-(3,1,5))=0~\implies~\boxed{5x+6y+8z-61=0}$