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Homework Help: Help Needed Solving For Two Scalars Such That au + bw = v

  1. Jan 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Find a and b such that v= au + bw given that v = <1, 1>

    2. Relevant equations
    v = au + bw, v = <1,1>

    3. The attempt at a solution
    I have tried to solve for the appropriate values via guess and check, but it hasn't worked out. I cannot find a proportion either through fractional or integer values that satisfy the equation.
  2. jcsd
  3. Jan 26, 2010 #2
    couldnt a be (1,1)/u???
  4. Jan 26, 2010 #3
    I am not quite sure what you mean. The values of a and b must be scalars.
  5. Jan 26, 2010 #4
    to be honest i've never seen the <> notation but i assume that means a vector. But how can you add two scalars to get a vector?
  6. Jan 26, 2010 #5
    This particular notation denotes a vector in component form. The initial point is the origin. The point given indicates the terminal point.
  7. Jan 26, 2010 #6
    Is there a generalized form that I could use to solve for two different scalars? Every system of equations I try to create is fundamentally flawed in some way.
  8. Jan 28, 2010 #7
    Heres a generalized problem that might help. Just plug and chug your numbers. You should be provided with u and w values.

    Heres my example
    v= <2,1> (here you'd put <1,1>. i just don't feel like resolving it) v=<1,2> (i+2j) and w=<-1,1> (i-j)

    therefore v=2i+j (<2,1> corresponds to the <i,j> coeffs)
    so set the two sides equal to eachother.
    2i + j = a(i + 2j) + b(i - j)
    2i + j = (a + b)i + (2a - b)j
    simplify and you get
    2=a+b and 1=2a-b
    Solve and you get a=1 and b=1

    If you need anymore help email me. :]
  9. Jan 29, 2010 #8


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    Science Advisor

    What are u and w? This makes no sense without knowing that. For some u and w, there will not exist such a and b. Assuming that u and w form a basis for R2, there must be unique a and b but they depend on what u and w are.

    If, for example, u= <1, 0> and w= <0, 1> then a= b= 1. If u= <1, 1> and v= <1, -1) then a= 1 and b= 0.

    If you are supposed to give some general formula, then let [itex]u= <u_1, u_2>[/itex] and [itex]w= <w_1, w_2>[/itex] then au+ bw= v becomes [itex]a<u_1, u_2>+ b<w_1, w_2>[/itex][itex]= <au_1+ bw_1, au_2+ bw_2>= <1, 1>[/itex] so we must have [itex]au_1+ bw_1= 1[/itex] and [itex]au_2+ bw_2= 1[/itex]. Solve those equations for a and b. Of course, the solution will involve a denominator of [itex]u_1w_2- u_2w_1[/itex]. There will be a solution if and only if that is not 0, exactly the condition that u and w form a basis.
    Last edited by a moderator: Jan 29, 2010
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