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Abstract vector basis

  1. Mar 31, 2015 #1
    1. The problem statement, all variables and given/known data
    Suppose that ## u = s_1i + s_2j ## and ## v = t_1i + t_2j ##, where s1, s2, t1 and t2 are real
    numbers. Find a necessary and sufficient condition on these real numbers
    such that every vector in the plane of i and j can be expressed as a linear
    combination of the vectors u and v.

    2. Relevant equations

    We shall need to consider directed line segments, and we denote the directed
    line segment from the point a to the point b by [a, b]. Specifically, [a, b] is the
    set of points {a + t(b − a) : 0 ≤ t ≤ 1},

    3. The attempt at a solution
    As attached.
     

    Attached Files:

  2. jcsd
  3. Mar 31, 2015 #2

    BvU

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    Phrase it this way: if ##\vec w = x \; \hat\imath + y \; \hat \jmath## then what do you have to do to write it as ## \vec w = a \; \vec u + b\; \vec v## ?
    When can you do that and when can you not do that ?
     
  4. Mar 31, 2015 #3
    ## \vec u = s_1 \vec i+ s_2 \vec j ##

    ## \vec v = t_1 \vec i+ t_2 \vec j ##

    ## x = a s_1 + b t_1 ##

    ## y = a s_2 + b t_2 ##
     
  5. Mar 31, 2015 #4

    Mark44

    Staff: Mentor

    Is it always possible to solve for a and b? What if u or v is the zero vector? What if u and v are equal? What if u is a nonzero multiple of v?

    You have a very simple space here -- the plane. The question boils down to this: what does it take for two vectors to span a plane?
     
  6. Apr 1, 2015 #5

    BvU

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    So, try to solve for a and b and see what you get !
     
  7. Apr 6, 2015 #6
    I believe I have only found two/three cases that restrict the coefficients of u and v. I have attached my solution.
     

    Attached Files:

  8. Apr 6, 2015 #7

    BvU

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    Solving for a and b means you express the unknowns a and b in terms of the knowns, x, y, s1, s2, t1 and t2.
     
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