Where's the error in this equation of a plane

Question

I was given the question:

Find equation of a plane with $P$$(-4,-4,-2)$ and normal vector $\langle -1,4,1 \rangle$.

My final answer was:

$$-x+4y+z=-10$$

But the last part is wrong $(-10)$.

How is it wrong? I believe is something small but I want to make sure I didn't get the formula wrong for this.

Answer 1

The equation of the plane satisfies $$r \cdot \begin{pmatrix} -1 \\ 4 \\ 1\end{pmatrix} = d$$

for some constant $d$ and any point $r$ on the plane. Substituting in $r = (-4, -4, -2)$ we get $$\begin{pmatrix}-4 \\ -4 \\ -2\end{pmatrix} \cdot \begin{pmatrix} -1 \\ 4 \\ 1\end{pmatrix} = 4 -16 - 2 = -14$$

Now using $r = \begin{pmatrix} x \\ y \\ z\end{pmatrix}$ we get $$-x + 4y + z = -14$$

Answer 2

The equation for a plane is implicitly given by $$\langle n,p-p_0 \rangle=0$$ where $\langle. ,.\rangle$ represents the dot product, $n$ is the normal vector, $p=(x,y,z)$, and $p_0$ is a point on the plane.

Put $n=(-1,4,1),p_0=(-4,-4,-2)$. We compute the inner product :

$$\langle (-1,4,1), (x+4,y+4,z+2) \rangle$$ to get :

$$-(x+4)+4(y+4)+(z+2)$$

Setting this equal to $0$, and moving the constants to the right side,

we find that $$-x+4y+z=4-16-2$$

so the equation becomes

$$-x+4y+z=-14$$