A
sphere is the set of all points in space equidistant from a fixed point, the center of the sphere (
[link] ), just as the set of all points in a plane that are equidistant from the center represents a circle. In a sphere, as in a circle, the distance from the center to a point on the sphere is called the
radius .
The equation of a circle is derived using the distance formula in two dimensions. In the same way, the equation of a sphere is based on the three-dimensional formula for distance.
Rule: equation of a sphere
The sphere with center
$\left(a,b,c\right)$ and radius
$r$ can be represented by the equation
Let
$P=\left(\mathrm{-5},2,3\right)$ and
$Q=\left(3,4,\mathrm{-1}\right),$ and suppose line segment
$PQ$ forms the diameter of a sphere (
[link] ). Find the equation of the sphere.
Since
$PQ$ is a diameter of the sphere, we know the center of the sphere is the midpoint of
$PQ.$ Then,
Describe the set of points in three-dimensional space that satisfies
${\left(x-2\right)}^{2}+{\left(y-1\right)}^{2}=4,$ and graph the set.
The
x - and
y -coordinates form a circle in the
xy -plane of radius
$2,$ centered at
$\left(2,1\right).$ Since there is no restriction on the
z -coordinate, the three-dimensional result is a circular cylinder of radius
$2$ centered on the line with
$x=2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=1.$ The cylinder extends indefinitely in the
z -direction (
[link] ).
Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows). With a three-dimensional vector, we use a three-dimensional arrow.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?