### Instruction: Let $\mathbf{w} = \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix}.$  The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \mathbf{0}\]lie on a plane.  Enter the equation of this plane in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$.### Response: The condition $\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \mathbf{0}$ means that the vector $\mathbf{v}$ is orthogonal to $\mathbf{w}.$
That is, their dot product is zero: $\mathbf{v} \cdot \mathbf{w} = 0.$
If I write $\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix},$ then this condition becomes
\[x \cdot 1 + y \cdot 0 + z \cdot (-3) = 0.\]
This is a linear equation in $x, y, z$ that defines a plane.
To write it in the standard form, I just need to rearrange the terms and simplify the coefficients.
I get $x - 3z = 0.$
This is already in the required form, where $A = 1,$ $B = 0,$ $C = -3,$ and $D = 0.$