Imagine three unit spheres (radius equal to 1) with centers...

Imagine three unit spheres (radius equal to 1) with centers at O(0,0,0), P(?3, -1,0), and Q(?3, 1,0). Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure). a. Find the coordinates of R. b. Let r_ij be the vector from the center of sphere i to the center of sphere j. Find r_OP, r_OQ, r_PQ, r_OR, and r_PR. a. Point R is at ( , , ). ( Type exact answers, using radicals as needed.)

Transcript

00:01 So in this question we have the following setup.

00:03 We've got x, y, and z.

00:06 So z, x, y.

00:09 And we've got some spheres.

00:10 So we've got sphere one centered at o.

00:16 We've got sphere two, which is centered at q.

00:22 Sphere 3 is centered at p.

00:25 And these are all spheres.

00:34 This is a sphere as well.

00:38 Then we put another sphere on top of them.

00:42 So this is going to be kind of hard to draw.

00:45 But here's another sphere on top of them with center r.

00:50 And if you're a bit confused by my drawing, have a look at the drawing that was posted along with this question.

00:56 It makes it a lot more clear.

00:58 So o is 0 -0.

01:01 P is root 3 minus 1 -0.

01:05 And q is root 3 1 ,0.

01:12 And we want to find where is r? well, we have a symmetry when y goes to minus y for the whole system.

01:26 And r obeys that symmetry.

01:31 So that means that ry is equal to r of minus y.

01:39 And that means that twice ry is equal to 0, so we can say that r is rx, 0, rz.

01:52 So we've already narrowed it down, that the y coordinates of r must be zero.

01:59 So now we can go about trying to work out the z and the x coordinates of r.

02:08 Well, we know that the sphere touches all three of the other.

02:14 The spheres.

02:17 So let's try and work out what that means.

02:21 Well, what it means is that the distance between p and r, q and r, and r and o and r are all the same, and they're all equal to twice the radius of the sphere.

02:31 So the distance from o to r is equal to the distance from p to r, is equal to the distance from q to r, which is equal to twice the radius of the sphere.

02:44 So this should give us enough equations to solve for the other two coordinates.

02:50 I mean, it gives us three equations, which definitely should be enough.

02:55 So the distance from o to r is rx squared plus rz squared, because ry squared is 0.

03:03 And actually we can square all of these, just to make the calculation easier.

03:09 So rx squared plus rz squared is equal to the distance from p to r, which is root 3 plus minus r x squared plus rz squared and this is equal to the distance from q to r which is again root 3 minus so actually but we need to include the one the one for y so this is the y distance and then this is the same and so you can see that we've already solved for the y coordinate means that these two equations are the same.

03:55 So we're left with rx squared plus rz squared is equal to 1 plus root 3 minus rx squared plus rz squared and that's equal to four.

04:09 So what we should do is we should have a look at this equality first and this tells us that rx squared is 1 plus root 3 minus rx squared.

04:25 So this is 1 plus 3 minus 2 root 3 rx plus rx squared...

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