Q.Find the Cartesian equation of the plane passing through A( -1, 2, 3), the direction ratios of whose normal are 0, 2, 5.

[!TLDR] Use $a(x-x_1)+b(y-y_1)+c(z-z_1)=0$ with $(x_1,y_1,z_1)=A$ and $(a,b,c)$ the given direction …

The plane passes through $A(-1,2,3)$ and its normal has direction ratios $0, 2, 5$.

By Theorem 6.9, the Cartesian equation is $a(x-x_1)+b(y-y_1)+c(z-z_1)=0$ with $(x_1,y_1,z_1)=(-1,2,3)$ and $(a,b,c)=(0,2,5)$: …

1. Read off the point $(x_1,y_1,z_1)$ and the direction ratios $(a,b,c)$ of the normal.
2. Substitute into $a(x-x_1)+b(y-y_1)+c(z-z_1)=0$. …

Since $a=0$, forgetting the $x$-term should genuinely vanish and instead accidentally including an $x$-term; also a sign slip on $x_1=-1$, writing $(x-1)$ instead of $(x+1)$ (which is …