10.5#11

Following the book, I believe that I have correctly solved for the plane's
equation. The normal vector is expressed as <-2,4,1> and following the book
my specific points should lead to -2(x--4)+4(y--3)+1(z-2)=0. Am I correct
in this assumption or am I missing something?

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Nope, not correct. A parametric equation for a line looks like r(t)=u+tv, where the vector u just gives a point on the line and the vector v determines the direction of the line: the direction of the line is read from the coefficients of t.   Instead you've used the u.