>> Suppose we want to find the equation of the sphere if the points, this point, and this point are endpoints of a diameter. So these are the endpoints of a diameter, and we'd like to write the equation. Hmm, how can we do that? Well gee, a diameter though goes through the center. And so if we can find the midpoint between these endpoints of the diameter, then we'll have the coordinates for the center. Now what about the radius? Well a strategy for finding radius is to take the coordinates for the center and the coordinates for either of those two endpoints, and just find the distance between those. Okay, let's do that. Now we need to keep things organized, so we might label like this. We might say okay, let's calculate the coordinates for the center. Now the center would be the midpoint between these two points. And remember, midpoint is the average of the Xs, average of the Ys, average of the Zs, or it's X sub 1 plus X sub 2 over 2, Y sub 1 plus Y sub 2 over 2, Z plus Z over 2. All right, and just plug in the values, and we find the coordinates of the center to be 1 1 4. All right, now we know where the center is, now what about the radius. Well we can find that radius distance by finding the distance between this point and either of the other two. Now we could also find it by taking the distance between those two and dividing it in half, that would be okay as well. But let's go directly at it, let's find the distance between this one and either of those two. All right, so for the radius, use the distance formula. Notice Delta X squared plus Delta Y squared, plus Delta Z squared, all under the radical. So now difference in X components. Well I'm using the first given point over here, the first coordinate was 3, and the coordinate for the center, the X coordinate for the center was 1. So 3 minus 1. And then from over there, negative 2, and then minus the 1. And then the 6 was the Y coordinate of that given point, and then minus the 4, the Y coordinate for the center. And then performing the calculations, we find the R value, the radius to be the square root of 17. And now all we do is put all of this together into the equation format. Here's the form of the equation. Now the H K L, those are the coordinates for the center, and those coordinates were 1 1 4. Now notice that those numbers follow minus signs here, that was an important consideration when we were talking about circles in earlier studies. But at any rate, so we have this part on the left, and then the R squared means the square root of 17 squared. But when we square the square root of 17, we get 17, and we would probably write this as 17 on the right. So here then is the equation we're looking for.