Question

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)?