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In summary, say that if ##\vec X## is defined to be an arbitrary vector centered on the origin, and R is a positive real number, then the set ##S_R=\{\vec{X}:X=R \}## would be the set of all vectors that point to the surface of the sphere radius R, centered on the origin. Thus - we could say that S_R "describes" a sphere radius R. ##S_1## would be the set that describes the unit sphere.
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## Homework Statement

Copy-paste from my textbook:

Let S_1 be the sphere of radius 1, centered at the origin. Let a be a
number > 0. If X is a point of the sphere S_1, then aX is a point of the sphere of radius a, because
||aX|| = a||X|| = a. In this manner, we get all points of the sphere of radius a. (Proof?)

## The Attempt at a Solution

On another site I posted this below:

Suppose we have a sphere S of radius 1 centered at the origin. Let X be a point on S. Then ||X - 0|| = 1.

Since ||cA|| = c||A|| for any vector A and c > 0, we have ||cX|| = c||X|| =c that is if we stretch the vector X by a factor of c, then the length stretches also by that amount. So, cX is a point on a sphere S_2 of radius c.

How do we show all the points of Sphere S_2 of radius c are cX?

You have S1={ |X|=1 }, S2={ |X|=c }, and cS1 = { cX for some X in S1 }, and you want to show S2 = cS1. You show X in S2 implies X in cS1 and vice-versa.

If X in S2, then |X|=c, and |(1/c)X|=(1/c)c=1, so (1/c)X is in S1, and X=c((1/c)X) is in cS1.

The other way, starting with X in cS1, so X=cY for some Y with Y in S1, then |X|=|cY|=c|Y|=c*1=c, so X in S2.

Are we showing that if cS1 equals S2, then cS1 is a sphere of radius c and since |cX| = c|X| = c, cX is a point on cS1?

Didn't get any answer. At this point I am very confused and have no idea what's going on. Can anyone please elaborate on this problem?

Thanks.

Lets see if I understand you - by rephrasing what you wrote:

If ##\vec X## is defined to be an arbitrary vector centered on the origin,
and R is a positive real number,
then the set ##S_R=\{\vec{X}:X=R \}## would be the set of all vectors that point to the surface of the sphere radius R, centered on the origin.

Thus - we could say that S_R "describes" a sphere radius R.

##S_1## would be the set that describes the unit sphere.

You want to know if you have managed to prove that ##RS_1=S_R##

Is this correct?

Simon Bridge said:
Is this correct?

Yes. Thank you.

Somehow managed to understand this :)

## 1. How does stretching a sphere by a radius affect its surface area?

Stretching a sphere by a radius increases its surface area by approximately 4πr^2, where r is the original radius of the sphere.

## 2. Does the volume of a sphere change when it is stretched by a radius?

No, stretching a sphere by a radius does not change its volume. The volume of a sphere is calculated using the formula (4/3)πr^3, which is independent of the radius.

## 3. What happens to the shape of a sphere when it is stretched by a radius?

Stretching a sphere by a radius will result in a prolate spheroid shape, where the length of the sphere's new radius is greater than its original radius. The shape of the sphere becomes more elongated as the radius is stretched further.

## 4. Is there a limit to how much a sphere can be stretched by a radius?

Yes, there is a limit to how much a sphere can be stretched by a radius. This limit is known as the critical radius, and it is equal to half of the original radius. Beyond this point, the sphere will no longer maintain its spherical shape.

## 5. What are the practical applications of stretching a sphere by a radius?

Stretching a sphere by a radius has various practical applications, such as in the manufacturing of lenses, mirrors, and other optical components. It is also used in geodesy for mapping the Earth's surface and in computer graphics for creating 3D models of objects.

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