Thought someone might enjoy this:(adsbygoogle = window.adsbygoogle || []).push({});

I'm working through my multivariable calc textbook but got stumped for a bit by some fusing algebra (duh). I finally saw the light this morning when I drew a 2D version on paper. Here's the problem:

We are given three vectors,r,a, andbwhere

r= <x, y, z>

a= <a1, a2, a3>

b= <b1, b2, b3>

Show that the equation (r-a) . (r-b) = 0 (dot product) defines a sphere. Find its radius and its center coordinates.

OK - that seemed pretty straightforward. But, when I plugged the components into the equation I came up with a rather ugly equation that didn't seem to make sense for a sphere nor did it provide me with much insight into the nature of the two vectorsaandb.

When I drew the problem (reducing things down to 2D) it wasn't hard to see how it was possible foraandbto determine a sphere and satisfy the dot product condition. Then it was easy to find the center and radius. A not-as-ugly equation for the sphere was then a cinch but it was still ugly enough (to me, anyway) to conceal any obvious equality to the first one. I ended up cheating and used Mathematica to prove that they really were equal :yuck: .

Can you solve this and understand the nature ofaandbstrictly from algebraic manipulation of the ugly version that you get when you insert the vectors' components into the dot product equation or do you have to make a drawing too?

Perion

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A 3D vector problem

**Physics Forums | Science Articles, Homework Help, Discussion**