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Multiplying vectors

  1. Jan 7, 2008 #1
    Three vectors have magnitudes a=3.00m (on the positive x-axis), b=4.00m (makes an angle of 30 degrees with the positive x-axis, counter-clockwise), and c=10.0m (makes an angle of 60 degrees with the negative x-axis, clockwise). If [tex]\overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}[/tex] what are the values of p and q?

    I know that the dot product is

    [tex]\overrightarrow{c}=\overrightarrow{a} \cdot \overrightarrow{b}[/tex]

    So...

    [tex]\overrightarrow{a} \cdot \overrightarrow{b}=p\overrightarrow{a}+q\overrightarrow{b}[/tex]

    I'm not really sure what my next step should be.
     
    Last edited: Jan 7, 2008
  2. jcsd
  3. Jan 7, 2008 #2

    berkeman

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    Staff: Mentor

    Hint: Draw the 3 vectors, with the head of a touching the tail of b, and the tail of c touching the tail of a. Do you see that you could ratio the sizes of a and b (remember that p & q can be negative numbers if needed), to make the head of the sum of pa+qb meet the head of c?
     
  4. Jan 8, 2008 #3
    Those are not even mathematically valid equations! In the first one, you have a vector on the left and a scalar on the right. (Remember, the dot product of two vector results in a scalar.) In the second, it's the other way around.

    The simplest way, if you ask me, is to equate components and solve the resulting set of linear equations.
     
  5. Jan 8, 2008 #4

    D H

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    Staff Emeritus
    Science Advisor

    As neutrino noted, your approach is not valid. You are on the right track, however. Taking the inner product with [itex]\vec a[/itex] and [itex]\vec b[/itex] yields a pair of linear equations in [itex]p[/itex] and [itex]q[/itex]:

    [tex]
    \begin{aligned}
    \vec a \cdot \vec c &= \vec a \cdot \vec a \, p + \vec a \cdot \vec b \, q \\
    \vec b \cdot \vec c &= \vec b \cdot \vec a \, p + \vec b \cdot \vec b \, q
    \end{aligned}
    [/tex]

    You have the requisite information needed to determine all of the inner products in the above equations. Solve the system of equations for [itex]p[/itex] and [itex]q[/itex], and voila, you have the answer.
     
  6. Jan 8, 2008 #5
    Can't believe I made such a dumb mistake. Alright, I will attempt this problem again with your suggestions, thanks!
     
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