Final answer:
The center of the sphere described by the equation where · means the dot product, and given vectors a = h 3, 5, 3 i and b = h 5, 9, 7 i, is the midpoint between points a and b, which is the point (4, 7, 5).
Step-by-step explanation:
To determine the center of the sphere when the equation (r - a)·(r - b) = 0 is given and vectors a and b are known, we need to realize that this equation represents the set of all vectors r that are equidistant from a and b, since the dot product is zero only when r - a and r - b are orthogonal, which is the case when r describes a sphere whose diameter ends are points a and b.
Given a = h 3, 5, 3 i, and b = h 5, 9, 7 i, the center of the sphere is the midpoint between a and b. To find it, we simply take the average of the corresponding components of vectors a and b.
- Center x-coordinate = (3 + 5) / 2 = 4
- Center y-coordinate = (5 + 9) / 2 = 7
- Center z-coordinate = (3 + 7) / 2 = 5
Therefore, the center of the sphere is the point (4, 7, 5).