- #1
Perion
Thought someone might enjoy this:
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, and b where
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 vectors a and b.
When I drew the problem (reducing things down to 2D) it wasn't hard to see how it was possible for a and b to 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 of a and b strictly 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
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, and b where
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 vectors a and b.
When I drew the problem (reducing things down to 2D) it wasn't hard to see how it was possible for a and b to 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 of a and b strictly 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
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