In this problem, we use the ideas of this section to derive a formula for the distance

Chapter 5, Problem 28

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In this problem, we use the ideas of this section to derive a formula for the distance from a point P(x0, y0,z0) in R3 to a plane P with equation ax + by + cz + d = 0. Unlike the situation of the distance from a point to a line in which we performed orthogonal projection onto a vector pointing along the line followed by vector subtraction, here we can project directly onto a normal vector n = (a, b, c) for the plane without the need for vector subtraction. (a) Draw a picture of this situation, including the point P, the plane P, and the normal vector n to the plane. (b) Choosing a point Q(x1, y1,z1) on the plane, construct the vector w from Q to P and show geometrically that the distance from P(x0, y0,z0) to the plane is ||P(w, n)||. (c) Find a formula for ||P(w, n)|| in terms of a, b, c, d, x0, y0, and z0.