   Chapter 12.3, Problem 54E

Chapter
Section
Textbook Problem

# If r = ⟨x, y, z⟩ a = ⟨a1, a2, a3⟩, and b = ⟨b1, b2, b3⟩, show that the vector equation (r − a) · (r − b) = 0 represents a sphere, and find its center and radius.

To determine

To find: The center and radius of the sphere.

Explanation

Given vector equation is (ra)(rb)=0 that is (ra) and (rb) vectors are orthogonal.

The cross-sectional view of the sphere is shown in Figure 1 with A, B and R terminal points of the vectors.

As the R lies on a sphere, whose diameter is the line from A to B. The center of the sphere is the midpoint of AB.

Center of the sphere is derived as follows.

12(a+b)=12(a1,a2,a3+b1,b2,b3)=12(a1+b1),(a2+b2),(a3+b3)=12(a1+b1),12(a2+b2),12(a3+b3)

Thus, the center of the sphere is 12(a+b)=12(a1+b1),12(a2+b2),12(a3+b3)_

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