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Homework Help: About stretching a sphere by a radius of a.

  1. Mar 14, 2014 #1
    1. The problem statement, all variables and given/known data

    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?)

    2. Relevant equations
    3. 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?

    I got this answer:

    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.

    Then I asked:

    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 whats going on. Can anyone please elaborate on this problem?

  2. jcsd
  3. Mar 14, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    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?
  4. Mar 16, 2014 #3
    Yes. Thank you.

    Somehow managed to understand this :)
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