Distance from a point to a planea. Show that the point in

Chapter 12, Problem 86AE

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Distance from a point to a plane

(a) Show that the point in the plane ax+by+cz=d nearest the origin is \(P\left(a d / D^{2}, b d / D^{2}, c d / D^{2}\right)\), where \(D^{2}=a^{2}+b^{2}+c^{2}\). Conclude that the least distance from the plane to the origin is |d|/D. (Hint: The least distance is along a normal to the plane.)

(b) Show that the least distance from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) to the plane ax+by+cz=d is \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\).

(Hint: Find the point P on the plane closest to \(P_{0^{-}}\))