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Math Help - Is this a Valid proof? [Vectors]

  1. #1
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    Is this a Valid proof? [Vectors]

    Suppose that a, b ∈ ℝ3 and a b = 0 Prove that there exist scalars λ, μ at least one of
    which is non-zero, such that \lambda a= \mu b.

    Suppose we have λ, μ such that μ does not equal zero. And that a and b are vectors.

    Using λa = μb we can re-arrange to get (λ/μ)a = b

    Plugging this into a x b = 0, we get;

    a x ( a(λ/μ)) = 0, re-arranging we get.

    (a x a) (λ/μ) = 0

    ((a x a)/μ) = 0


    Am i right so far? / Can any 1 show me the proof ?
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  2. #2
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    Quote Originally Posted by simpleas123 View Post
    Suppose that a, b ∈ ℝ3 and a b = 0 Prove that there exist scalars λ, μ at least one of
    which is non-zero, such that \lambda a= \mu b.
    I don't see where you are going with that.

    If either a \text{ or }b is the zero vector you are done.
    So suppose neither is the zero vector.
    If \phi is the angle between them then \sin (\phi ) = \frac{{\left\| {a \times b} \right\|}}{{\left\| a \right\|\left\| b \right\|}}.
    From the given, what does that tell you about \phi?
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  3. #3
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    Quote Originally Posted by Plato View Post
    I don't see where you are going with that.

    If either a \text{ or }b is the zero vector you are done.
    So suppose neither is the zero vector.
    If \phi is the angle between them then \sin (\phi ) = \frac{{\left\| {a \times b} \right\|}}{{\left\| a \right\|\left\| b \right\|}}.
    From the given, what does that tell you about \phi?
    But dont we have to prove that either Lander or Mew is zero?
    And it does it mean Theta = 180?
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  4. #4
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    Quote Originally Posted by simpleas123 View Post
    But dont we have to prove that either Lander or Mew is zero?
    And it does it mean Theta = 180?
    Actually it means that \phi=0\text{ or }\phi=\pi.
    In turn that means that a~||~b or one is a multiple of the other.
    Just recall that neither is the zero vector.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Actually it means that \phi=0\text{ or }\phi=\pi.
    In turn that means that a~||~b or one is a multiple of the other.
    Just recall that neither is the zero vector.
    Right i sort of understand it, so how would i write out the full proof?
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