1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?

    Thanks.
     
  2. jcsd
  3. Mar 14, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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 :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: About stretching a sphere by a radius of a.
  1. Radius of a circle (Replies: 10)

  2. Radius of a circle (Replies: 1)

  3. Radius of Circle (Replies: 2)

  4. Radius of the track (Replies: 1)

Loading...