Calculus III:
Find an equation of a sphere if one of its diameters has endpoints
(3, 5, 6) and (5, 7, 8).

Question

Calculus III: 
Find an equation of a sphere if one of its diameters has endpoints 
(3, 5, 6) and (5, 7, 8).

misty1212 · Answer

HI!!
this is not as hard as it looks

misty1212 · Answer

find the center first, the average in each coordinate

misty1212 · Answer

you got that?

anonymous · Answer

anticipation is killing me...

k8lyn911 · Answer

Sorry! Working through some other homework problems.
Which equations do we use for that?

misty1212 · Answer

just like the midpoint in two dimensions 
take the average in each coordinate

k8lyn911 · Answer

So (x1 - x2, y1 - y2, z1 - z2)?

misty1212 · Answer

so, the average

misty1212 · Answer

add up and divide by two

misty1212 · Answer

"no" i meant 
just like finding the midpoint of $(3,5)$ and $(7,11)$ in two dimensions 
add and divide by two in each coordinate 
in my example the midpoint would be $(5,8)$

misty1212 · Answer

clear or  no?

k8lyn911 · Answer

( (3+5)/2, (5+7)/2, (6+8)/2) = (4, 6, 7)?

misty1212 · Answer

right

misty1212 · Answer

before we continue, do you remember how to find the equation of a circle given two endpoints of the diameter? this is identical, except with one extra coordinate

misty1212 · Answer

the equation is going to be $$(x-4)^2+(y-6)^2+(z-7)^2=r^2$$

k8lyn911 · Answer

Not exactly...

misty1212 · Answer

because the center is $(4,6,7)$

misty1212 · Answer

equation for a circle with radius $r$ and center $(h,k)$ is 
$$(x-h)^2+(y-k)^2=r^2$$

misty1212 · Answer

this is exactly the same, but with one more coordinate

k8lyn911 · Answer

Oh, so it's like the other problems we've had, but backwards!
So we add these values squared and take the square root to find r?

misty1212 · Answer

ok lets go slow

misty1212 · Answer

maybe i am not sure what exactly you mean 
we got the center, we need the radius

misty1212 · Answer

actually we don't need the radius, we just need the square of the radius, so forget the square root business

k8lyn911 · Answer

Okay.

misty1212 · Answer

same distance formula as with two points, only now each point has three coordinates instead of two

misty1212 · Answer

square of the distance between $(3, 5, 6) $ and $ (4, 6, 7)$ is 
$$((4-3)^2+(6-5)^2+(7-6)^2$$ a pretty easy calculation in this case

misty1212 · Answer

for the actual distance it would be $$\sqrt{(4-3)^2+(6-5)^2+(7-6)^2}$$

misty1212 · Answer

you get 3 pretty much instantly right?

k8lyn911 · Answer

Yes.

misty1212 · Answer

final answer:$$(x-4)^2+(y-6)^2+(z-7)^2=3$$

misty1212 · Answer

a nice picture http://www.wolframalpha.com/input/?i=%28x-4%29%5E2%2B%28y-6%29%5E2%2B%28z-7%29%5E2%3D3

k8lyn911 · Answer

Oh, I see.
Thank you so much for explaining it to me! :)

misty1212 · Answer

$$\huge \color\magenta\heartsuit$$