Vectory Geometry, problem involving spheres

[josephcollins]

Hi ppl, ne1 know how to approach this one?

Given P(1,4,-1), Q(1,2,-1) and R (3,2,-2), show that PQR is right angled and hence find the equation of the smallest sphere S which passes through P,Q and R.

Okay, I got the right angle by showing PQ and QR to be perpendicular with the dot product as zero.
With the equation of the sphere, I'm not sure how to start this, could someone just point me in the correct direction,
Thanks,
Joe

[Alkatran]

Hi ppl, ne1 know how to approach this one?

Given P(1,4,-1), Q(1,2,-1) and R (3,2,-2), show that PQR is right angled and hence find the equation of the smallest sphere S which passes through P,Q and R.

Okay, I got the right angle by showing PQ and QR to be perpendicular with the dot product as zero.
With the equation of the sphere, I'm not sure how to start this, could someone just point me in the correct direction,
Thanks,
Joe

You know that the center of the sphere is the same distance from all the points.

So: sqr((x - 1)^2 + (y - 4)^2 + (z - -1)^2) = sqr((x -1 ....

[mathman]

Alkatran's answer will give you two equations in three unknowns, since it holds for all spheres. To get the smallest sphere, you need the condition that the center lies in the plane determined by the three points. A way of approaching it, is to work in that plane, and get a circle passing through the three points. The sphere you want will have that circle as the equator.