Chapter 12.3, Problem 54E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

If **r** = ⟨*x*, *y*, *z*⟩ **a** = ⟨*a*_{1}, *a*_{2}, *a*_{3}⟩, and **b** = ⟨*b*_{1}, *b*_{2}, *b*_{3}⟩, 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

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.

Thus, the center of the sphere is

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