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A conceptual question on vectors

  1. Apr 4, 2005 #1
    Q. Can the magnitude of the difference of two vectors be ever be greater than the sum of the magnitudes of the two?

    Any insights, or answers shall be appreciated....
    Thanks
     
  2. jcsd
  3. Apr 4, 2005 #2

    dextercioby

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    Consider the case when the angle between the 2 vectors is in the interval [itex] \left(\frac{\pi}{2},\pi\right) [/tex]...What happens then...?

    Daniel.

    EDIT:Nothing happens.See posts #5 & #6.
     
    Last edited: Apr 4, 2005
  4. Apr 4, 2005 #3
    The answer is no, but, of course, you should ask why!

    Draw a triangle, with sides [itex]a, b, c[/itex] and with angles, opposite to the respective sides, [itex]A, B, C[/itex]. Now, remember way back in the past when you learned the cosine law? To remind you, it says that for such a triangle,

    [tex]c^2 = a^2 + b^2 - 2ab\cos C.[/tex]

    Now, see if that helps at all :smile:

    Actually drawing the triangle is advisable!
     
    Last edited: Apr 4, 2005
  5. Apr 4, 2005 #4

    dextercioby

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    The answer is YES for obtuse triangles ([itex] C\in \left(\frac{\pi}{2},\pi\right) [/itex]) just as i suggested above...:wink:

    Daniel.

    EDIT:See posts #5 & #6.
     
    Last edited: Apr 4, 2005
  6. Apr 4, 2005 #5
    No it's not~

    http://mathworld.wolfram.com/TriangleInequality.html

    dexter forgetting things in your old age! :biggrin:

    What we want is

    [tex]\| x - y \| \leq \|x\| + \|y\|[/tex]

    the triangle inequality is

    [tex]\| x + y \| \leq \|x\| + \|y\|[/tex]

    replace [itex]y[/itex] with [itex]-y[/itex] to obtain

    [tex]\| x - y \| \leq \|x\| + \|-y\|[/tex]

    but [itex]\|-y\| = \|y\|[/itex] so

    [tex] \|x - y\| \leq \|x\| + \|y\|[/tex]

    as we wanted.

    I proved the Cauchy-Schwarz inequality, which leads to the triangle inequality, the other day:

    https://www.physicsforums.com/showthread.php?t=69574
     
    Last edited: Apr 4, 2005
  7. Apr 4, 2005 #6

    dextercioby

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    Yes,you're right.

    Daniel.
     
  8. Apr 4, 2005 #7
    I think people are forgetting everything lately... I've made several silly mistakes in the past few days. Probably all that air pollution these days :wink:
     
  9. Apr 4, 2005 #8

    dextercioby

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    Bulls***.It's human nature.It reminds us that u're infallible...:wink:

    Where would all the fun be,if everyone was perfect/right all the time...?

    Daniel.

    P.S.I know this post was a lame excuse...:tongue2:
     
  10. Apr 4, 2005 #9
    I don't need to be reminded that I'm infallible! :tongue:
     
  11. Apr 4, 2005 #10

    dextercioby

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    Freudian slip??No,i sometimes forget the "no,not" all those negations...Of course i meant just the opposite...You see,errors make the fun in life...:tongue2:

    Daniel.
     
  12. Apr 4, 2005 #11

    HallsofIvy

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    A straight line is the shortest distance between 2 points: The sum of lengths of 2 sides of a triangle cannot be less than the length of the third side. That answers your question.
     
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