Define a perpendicular plane to the vector

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I want to define a perpendicular plane to the vector (2, 3, 4). The plane passes through the origin. And I want to define arbitrary points on this plane.

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The plane is $\pi:\ 2x+3y+4z=0$. It passes through the origin clearly, and its perpendicular to the line $L=span(2,3,4)$. This is because every vector $v\in \pi$ obviouly satisfies the equation of the plane, and so, when doing the dot product with the vector $(2,3,4)$ gives $0$, thus is orthogonal.

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STRONG HINT

Planes perpendicular to the vector $(2,3,4)$ have equation $$2x+3y+4z=c.$$ Can you now see what value the constant $c$ must take?