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Find the coordinates of a and b

  1. Feb 14, 2010 #1
    We are given two vectors [tex]\vec{a}[/tex] i [tex]\vec{b}[/tex]. We know that

    [tex]2\vec{a}-3\vec{b}=\begin{bmatrix}0\\-17\end{bmatrix}[/tex]

    [tex]\vec{a}+4\vec{b}=\begin{bmatrix}11\\19\end{bmatrix}[/tex]

    I have tried to solve this through a set of equations with two unknown, but don't really know how to do that:

    [tex]\begin{cases}2a-3b=[0,-17]\\a+4b=[11,19]\end{cases}[/tex]

    Thank you in advance for any clues:)
     
  2. jcsd
  3. Feb 15, 2010 #2
    As you can see from the original equations, the x and y components of the vectors are independent.
     
  4. Feb 15, 2010 #3
    Try to express what you know as a matrix equation: a matrix containing the components of the unknown vectors, multiplied by a matrix of some numbers equals another matrix of some numbers.
     
  5. Feb 15, 2010 #4

    HallsofIvy

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    You can solve just as you would any pair of equations, just being careful to keep both components on the right. For example, if I had "2a- 3b= " and "a+ 4b= " whether they were number, vectors, matrices, or whatever, I would think "If I multiply the second equation by 2 and subtract, I will eliminate the 'a'!"

    Okay, 2a- 3b= [0,-17] and 2a+ 8b= [22, 38]. Subtracting the first from the second, 11b= [22-0, 38+17]= [22, 55] and now I see that b= [2, 5].
     
  6. Feb 15, 2010 #5
    Oh, yeah, that's simpler than what I had in mind:

    [tex]\begin{bmatrix}a_1 & b_1\\a_2 & b_2\end{bmatrix}\begin{bmatrix}2 & 1\\-3 & 4\end{bmatrix}=\begin{bmatrix}0 & 11\\-17 & 19\end{bmatrix}[/tex]

    [tex]\begin{bmatrix}a_1 & b_1\\a_2 & b_2\end{bmatrix}\begin{bmatrix}2 & 1\\-3 & 4\end{bmatrix}\begin{bmatrix}2 & 1\\-3 & 4\end{bmatrix}^{-1}=\begin{bmatrix}0 & 11\\-17 & 19\end{bmatrix}\begin{bmatrix}2 & 1\\-3 & 4\end{bmatrix}^{-1}[/tex]

    [tex]\begin{bmatrix}a_1 & b_1\\a_2 & b_2\end{bmatrix}=\begin{bmatrix}0 & 11\\-17 & 19\end{bmatrix}\begin{bmatrix}\frac{4}{11} & -\frac{1}{11}\\\frac{3}{11} & \frac{2}{11}\end{bmatrix} = ...[/tex]
     
  7. Feb 15, 2010 #6

    Redbelly98

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    Moderator's note:
    Thread moved to Homework area. Homework assignments or any textbook style exercises are to be posted in the appropriate forum in our Homework & Coursework Questions area. This should be done whether the problem is part of one's assigned coursework or just independent study.
     
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