Results 1 to 12 of 12

Math Help - Complex vector question

  1. #1
    Newbie
    Joined
    Apr 2011
    Posts
    13

    Complex vector question

    Hi, here's a question from an exam that I don't know how to do. I can do (i) and (ii). For (ii) I get z_2-z_1 or z_1-z_2. But I don't know what they want for (iii) and (iv).

    I was thinking of using the fact that the diagonals of a paralleogram bisect each other, but that doesn't seem to work out...

    (i) Let z_1=r_1 cis \theta_1 and z_2=r_2 cis \theta_2. If z_1 and z_2 are parallel then prove that z_1 = k z_2 for k real.

    (ii) Let the points A, B, C and D be represented by the complex numbers z_1, z_2, z_3 and z_4 respectively (in clockwise order). Find two possible vectors representing side AB.

    (iii) If A, B, C and D are the vertices of a parallelogram then using parts (i) and (ii) find an expression for side DC in terms of AB using the vectors z_1, z_2, z_3 and z_4.

    (iv) Using a property of a parallelogram find the two possible values of k.

    (v) Show that for both these values of k that z_1-z_2-z_3+z_4=0.

    Any ideas?

    Thanks

    EDIT: Does this belong in the pre calculus section? If so, many apologies. My university considers complex numbers as an algebra topic...
    Last edited by minifhncc; May 7th 2011 at 05:25 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    Some of these problems seem strange.

    For (iii), the vector representing the side DC is equal to the vector sum DA + AB + BC, which I guess is an expression using AB and the four given vectors.

    (iv): Hm? We know that the two vectors z_1 and z_2 are parallel. Doesn't this mean that the points A and B lie on the same straight line through the origin? I don't see how this restricts the value of k in any way.
    Last edited by HappyJoe; May 7th 2011 at 05:18 AM. Reason: TeX
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2011
    Posts
    13
    Hello,

    Sorry I made a typo the line in the last part should be " z_1-z_2-z_3+z_4=0."

    Thanks again
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Apr 2011
    Posts
    13
    Quote Originally Posted by HappyJoe View Post
    Some of these problems seem strange.

    For (iii), the vector representing the side DC is equal to the vector sum DA + AB + BC, which I guess is an expression using AB and the four given vectors.

    (iv): Hm? We know that the two vectors z_1 and z_2 are parallel. Doesn't this mean that the points A and B lie on the same straight line through the origin? I don't see how this restricts the value of k in any way.
    I think it's hinting at the fact that sides DC and AB are parallel, and I need to use that result somehow...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    Quote Originally Posted by minifhncc View Post
    I think it's hinting at the fact that sides DC and AB are parallel, and I need to use that result somehow...
    As I see it, the vectors z_1 and z_2 don't have anything to do with the side DC, do they? They represent the point A and B, and for the two vectors to be parallel, the two points A and B have to lie on the same straight line through the origin.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Apr 2011
    Posts
    13
    I think it means to take it generally.

    ie. z_1 and z_2 in part (i) aren't the same vectors in the following parts.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by minifhncc View Post
    I think it means to take it generally.
    ie. z_1 and z_2 in part (i) aren't the same vectors in the following parts.
    From reply #2, “Some of these problems seem strange”. I absolutely agree with that. Moreover, whoever wrote certainly knows what he/she means. But that does not mean that we do. That said, here are some observations.

    If A:z_1,~ B:z_2,~ C:z_3,~\&~ D:z_4, are the vertices of a parallelogram in counter-clockwise order the side AB can be represented as the vector <\mathif{Re}(z_2)- \mathif{Re}(z_1), \mathif{Im}(z_2)- \mathif{Im}(z_1)>.

    Because we have a parallelogram, it must be true that  \mathif{Re}(z_3)- \mathif{Re}(z_4)= \mathif{Re}(z_2)- \mathif{Re}(z_1),~
    \&~ \mathif{Im}(z_3)- \mathif{Im}(z_4)= \mathif{Im}(z_2)- \mathif{Im}(z_1) .

    Does that help?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Apr 2011
    Posts
    13
    Quote Originally Posted by Plato View Post
    Because we have a parallelogram, it must be true that  \mathif{Re}(z_3)- \mathif{Re}(z_4)= \mathif{Re}(z_2)- \mathif{Re}(z_1),~
    \&~ \mathif{Im}(z_3)- \mathif{Im}(z_4)= \mathif{Im}(z_2)- \mathif{Im}(z_1) .

    Does that help?
    Yes I know that the diagonals of a parallelogram bisect each other... but how do I get the two values of k ...? Wouldn't that expression only give one value of k?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by minifhncc View Post
    Yes I know that the diagonals of a parallelogram bisect each other... but how do I get the two values of k ...? Wouldn't that expression only give one value of k?
    Well, my reply has absolutely nothing to do with the diagonals of a parallelogram.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    We want to find k in z_1=kz_2, is that right?

    In part (iv), what is your interpretation of z_1 and z_2, that is, which part of the parallelogram do they represent?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Apr 2011
    Posts
    13
    z_1 and z_2, I think, are just two arbitary vectors.

    ie. I'm thinking that since sides AB and DC are parallel, we need to find k such that z_3-z_4=k(z_2-z_1). Upon solving it I get k=\pm 1. But for both these values of k I don't get the required expression. I only get it for one value ie. k=-1...
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    Quote Originally Posted by minifhncc View Post
    z_1 and z_2, I think, are just two arbitary vectors.

    ie. I'm thinking that since sides AB and DC are parallel, we need to find k such that z_3-z_4=k(z_2-z_1). Upon solving it I get k=\pm 1. But for both these values of k I don't get the required expression. I only get it for one value ie. k=-1...
    This is what Plato said.

    You are right that for a pair of parallel vectors a and b, there exists only one k, such that a=kb, so them asking for two such values of k is strange.

    Unless of course they also want a k such that z_4-z_1=k(z_3-z_2), where we are looking at sides AD and BC, which they just might do, even though the problem text takes care not to give anything away.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. complex numbers and vector spaces
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 7th 2010, 12:58 PM
  2. Interpreting a zero norm of complex vector
    Posted in the Advanced Algebra Forum
    Replies: 14
    Last Post: February 12th 2010, 04:37 AM
  3. Complex Conjugation Vector Space
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 9th 2009, 07:07 PM
  4. complex numbers- vector- albebra
    Posted in the Math Topics Forum
    Replies: 4
    Last Post: June 29th 2008, 02:11 AM
  5. Replies: 3
    Last Post: June 1st 2008, 01:51 PM

Search Tags


/mathhelpforum @mathhelpforum