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Math Help - Vector Equality Proofs

  1. #1
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    Vector Equality Proofs

    I'm a bit confused about how to approach these questions. The only method that I have been using is trying to disprove options by counterexample to get to the solution.

    Under what condition is: |\vec A - \vec B| = A + B

    a) The magnitude of vector B is zero.

    |<-2, 7> - <0, 0>| = <-2, 7> + <0, 0 >
    sqrt(53) = sqrt(53)


    b) Vectors A and B are in the same direction.

    |<2, 0> - <3, 0>| = <2, 0> + <3, 0>
    1 != 13


    c) Vectors A and B are in perpendicular directions.

    |<2, 3> - <-1, 2>| = <2, 3> + <-1, 2>
    sqrt(10) != ~4.84

    d) Vectors A and B are in opposite directions.

    |<3, 0> - <-3, 0>| = <3, 0> + <-3, 0>
    6 = 6


    e) The statement is never true


    I'm very confused as to how to come to a definitive answer since a, d, and e are still in the game.

    And I have to do |\vec A - \vec B| = A - B next. :/
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  2. #2
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    Re: Vector Equality Proofs

    If you have given examples in which |A+ B|= |A|+ |B| the anyone of those is a counter-example to (e).

    But for (a), (b), (c), and (d) if you are trying to prove that under each of those conditions, |A+ B|= |A|+ |B|, an example is NOT a proof.
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  3. #3
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    Re: Vector Equality Proofs

    Quote Originally Posted by Algebrah View Post
    I'm a bit confused about how to approach these questions. The only method that I have been using is trying to disprove options by counterexample to get to the solution.

    Under what condition is: |\vec A - \vec B| = A + B
    Sorry to tell you but the notation |\vec A - \vec B| = A + B makes no sense.

    |\vec A - \vec B| is a nonnegative number (a scalar), the length of difference of two vectors.

    I don't know what you mean by  A + B. In what sense is that a nonnegative number?
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  4. #4
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    Re: Vector Equality Proofs

    I think that they want me to find the magnitudes of vectors A and B on the right side, then sum them to see if the scalar values are equivalent on each side. The problem is that I don't know how to do that without using specific chosen values.

    I agree that the right side of the equation is confusing, but that was how it was written in the book.
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  5. #5
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    Re: Vector Equality Proofs

    I'll probably have to try to use constants b, c, etc. in order to prove each one.
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  6. #6
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    Re: Vector Equality Proofs

    Quote Originally Posted by Algebrah View Post
    I think that they want me to find the magnitudes of vectors A and B on the right side, then sum them to see if the scalar values are equivalent on each side. The problem is that I don't know how to do that without using specific chosen values.

    I agree that the right side of the equation is confusing, but that was how it was written in the book.
    This is true: \|\vec{A}-\vec{B}\|\le \|\vec{A}\|+\|-\vec{B}\|=\|\vec{A}\|+\|\vec{B}\|.

    But it is easy to find a counter-example to equality.

    See if you can find such and post it here.
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  7. #7
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    Re: Vector Equality Proofs

    Quote Originally Posted by Algebrah View Post
    Under what condition is: |\vec A - \vec B| = A + B
    |\vec A - \vec B| = A + B iff (1) \vec{A} and \vec{B} are collinear and have opposite directions, or (2) one of the vectors is null. Depending on the definition, (2) may be a special case of (1); then only (1) is necessary.
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  8. #8
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    Re: Vector Equality Proofs

    Meh, I got both of them wrong. I'm just plain stumped. Thanks for the help everyone.
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