INSTRUCTOR: "Find the equation of the sphere with diameter line segment PQ, where P equals the ordered triple, 2 comma negative 1 comma negative 3, and Q equals ordered triple, negative 2 comma 5 comma negative 1." Now, we know that the equation of a sphere is given by this equation quantity x minus a squared plus quantity y minus b squared plus quantity z minus c squared equals r squared, where c, the center of the sphere, is the point a, b, c, and r is the radius of the sphere.

So first, what we need to do to complete that equation is we need to find what the center of our sphere is. Well, since we're given a diameter, we know that the center is the midpoint of that line segment. So we're going to average the components to find c.

c equals, and our ordered pair will begin with 2 plus negative 2 all over 2. That's our x component. Negative 1 plus 5 all divided by 2. That'd be our y component. And negative 3 plus negative 1 over 2. That'd be our z component.

That results in the point 0 comma 2 comma negative 2. So there's our center. Now, the only other piece of information we need is the radius of the sphere. There are two ways we could find this.

One is we could find the distance between the two points, the diameter, and then divide that by 2. Or since we know the center, we can find the distance from the center to one of the two endpoints. So I think I will choose to find the distance from the center to the point P, all right?

So let's see, the distance from P to c. I'll just notate it that way, d of P comma c. And that will be equal to the square root of, parentheses, 0 minus 2, close parentheses, squared plus, parentheses, 2 minus negative 1, close parentheses, squared plus, parentheses, negative 2 minus negative 3, close parentheses, squared.

Now, 0 minus 2 squared will be 4. 2 minus negative 1 squared. That would be 3 squared, 9. Negative 2 minus negative 3. Let's see, that would be positive 1 squared, so plus 1.

So we have the square root of 4 plus 9 plus 1. 4 plus 9 plus 1 is 14. So the distance is the square root of 14. That is our radius. So r equals square root of 14.

Now, fitting this into our equation, we have the equation is given by, parentheses-- well, actually, we don't need parentheses there. It'll just be x squared because the value of a is 0 plus, parentheses, y minus 2, close parentheses, squared plus, parentheses, z plus 2, close parentheses, squared equals 14 because the square of square root of 14 is 14. So that is the equation of our sphere given those two points in the diameter.