Results 1 to 5 of 5

Math Help - Linear Algebra problem

  1. #1
    Junior Member
    Joined
    Dec 2006
    Posts
    28

    Linear Algebra problem

    I was wondering how to go about a certain problem that confused me...

    Here's the question:

    Let x and y be nonzero vectors in the set of real n-vectors.

    Prove that the length(x+y) is equal to the length(x) + length(y) if and only if y = cx for some c greater than 0.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by faure72 View Post
    I was wondering how to go about a certain problem that confused me...

    Here's the question:

    Let x and y be nonzero vectors in the set of real n-vectors.

    Prove that the length(x+y) is equal to the length(x) + length(y) if and only if y = cx for some c greater than 0.
    Here is the idea.

    This is the triangular inequality.
    |x+y|<=|x|+|y|
    Meaning the triangle you form out of these vectors follows the rule that the sum of two sides of a triangle is greater than the third side.
    The exception (special case when we have equality) is when the vector, say "y" lies directly upon the other vector x. In that case we do not form a triangle and have equality.
    Thus, we need that y lies (coincides) with x. Meaning they have the same direction. And two vectors has the same direction iff they are non-zero scalar multiples of each other. Thus, y=cx
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2006
    Posts
    28
    Thanks! So I get the part about the triangular inequality. What did you mean, though, when you said "y lies directly upon the other vector x," though, when you were referring to the exception? Is it possible to show a picture of this?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by faure72 View Post
    Thanks! So I get the part about the triangular inequality. What did you mean, though, when you said "y lies directly upon the other vector x," though, when you were referring to the exception? Is it possible to show a picture of this?
    Heir.
    Attached Thumbnails Attached Thumbnails Linear Algebra problem-picture12.gif  
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Gee, I hate to do this without the use of TeX. Please try to follow the notation.
    If x=cy with c>0 then the angle between x & y is 0: thus x.y=|x||y|. (|V| is the length of V and V.W is the dot product of V with W.)
    Recall that |x+y|^2= |x|^2 + 2(x.y) + |y|^2.
    So if x.y=|x|y| then |x+y|^2= |x|^2 + 2|x||y| + |y|^2=(|x|+|y|)^2.
    Thus |x+y|=|x|+|y|.

    If |x+y|=|x|+|y| then |x+y|^2= |x|^2 + 2|x||y| + |y|^2.
    But |x+y|^2= |x|^2 + 2(x.y) + |y|^2 which implies (x.y)=|x||y| or x is positive multiple of y.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear Algebra Problem
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 16th 2009, 05:35 AM
  2. Linear Algebra Problem
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: July 19th 2009, 07:58 AM
  3. Linear Algebra Problem
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 4th 2009, 02:36 PM
  4. linear algebra problem
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 8th 2009, 05:23 PM
  5. another Linear Algebra problem
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 22nd 2007, 03:24 PM

Search Tags


/mathhelpforum @mathhelpforum