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Vector Proof Assignment

  1. Oct 30, 2006 #1
    Hi there, I had an assignment due today for my advanced maths class, I've already handed it in, but seeing what other people did I began to worry/wonder about what I did.

    The question was pretty simple:

    For non-zero vectors show that:
    |a - b| = |a + b| if and only if venctors a and b are perpendicular.

    I defined them both as different single letters, just to make things easier.
    a + b = c
    a - b = d

    Then I used the dot product on them both:

    c^2 = (a + b) o (a + b)
    c^2 = |a + b| |a + b| cos A
    c^2 = a^2 + b^2 + 2ab cos A

    In this case A is the angle opposite vector c, ie. the angle between vectors a and b.

    d^2 = (a - b) o (a - b)
    d^2 = |a - b| |a - b| cos B
    d^2 = a^2 + b^2 - 2ab cos B

    In this case B is the angle opposite vector d, ie. the angle between vectors a and -b.

    I then said that in order for |c| to equal |d| the following would have to be true:
    cos A = cos B = 0 because otherwise you would have to add/subtract.

    Of course it follows from this that A = B = 90 hence a and b are perpendicular if |c| = |d|

    I think it's pretty easy to see from that that if they're perpendicular then |c| = |d|, but my maths teachers don't take kindly to the "no duh" argument, they claim that's what proof is all about, so i proved it backwards to show the if and only if statement.

    My question is: other people did it using the dot product in a different way to me, does my method make sense? Also, should I have proved it backwards? I could lose marks for having irrelevant lines, but I could've lot lines for missing lines, it's a fine line.

    Worried about this because when I did it it was a really easy assignment and I don't want to mess it up :(


    Alright, I removed the tex formatting, as it was too confusing for me.
    Last edited: Oct 30, 2006
  2. jcsd
  3. Oct 30, 2006 #2


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    Staff Emeritus
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    Gold Member

    The way you did it looks good to me. It's a little confusing since you have more lines than are really necessary when showing what c2 and d2 are, but other than that, it looks fine.

    Yes, you should have proven it backwards, because it's an iff statement. Even if it looks obvious that one way implies the other, you should show it
  4. Oct 30, 2006 #3
  5. Nov 1, 2006 #4
    Well, thanks for your answers, didn't notice that other thread, that was useful too :smile:

    Incidently, I also got the second part of that question today.
    Last edited: Nov 1, 2006
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