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In $\mathbb{R}^{3}$, let $N$ be a non-zero vector and $X_{0}$ and $Z$ points.
a) Find the equation of the plane through the origin that is orthogonal to $N$, so $N$ is a normal vector to this plane.
b) Compute the distance from the point $Z$ to the origin.
c) Find the equation of the plane parallel to the above plane that passes through the point $X_{0}$.
d) Find the distance between the parallel planes in parts a) and c).
e) Let $S$ be the sphere centered at $Z$ with radius $r$. For which value(s) of $r$ is this sphere tangent to the plane in part c)?
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