Answer
An equation of the plane is
$2x + y - 4z = - 21$
Work Step by Step
By Theorem 1 of Section 13.5, the equation of a plane through ${P_0} = \left( {{x_0},{y_0},{z_0}} \right)$ with normal vector ${\bf{n}} = \left( {a,b,c} \right)$ is given by $ax + by + cz = d$, where $d = a{x_0} + b{y_0} + c{z_0}$. Therefore, the equation of the plane through $\left( {1, - 3,5} \right)$ with normal vector ${\bf{n}} = \left( {2,1, - 4} \right)$ is
$2x + y - 4z = d$,
where $d = 2\cdot1 + 1\cdot\left( { - 3} \right) - 4\cdot5 = - 21$.
Thus, $2x + y - 4z = - 21$.
