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Math Help - Vectors

  1. #1
    Member Veronica1999's Avatar
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    Vectors

    Give the components of the vector whose length is 10 and whose direction opposes the direction of [ -4, 3 ]

    Can I get some help pls?
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  2. #2
    Senior Member eumyang's Avatar
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    v_1 = <-4, 3>

    \theta_1 = \arctan \left( \frac{3}{-4} \right) + \pi \approx 2.498 \,rad
    (we add pi because theta is in the 2nd quadrant)

    \theta_2 = \theta_1 + \pi \approx 2.498 + \pi \approx 5.640 \,rad
    We want our new vector to be in the opposite direction, so add pi again.
    One could argue that adding pi twice isn't necessary, and finding arctan (3/-4) would give us the angle we're looking for. But I like angles to be in [0, 2\pi).

    Our new vector is
    \begin{aligned}<br />
v_2 &= <|v_2|\cos \theta_2, |v_2|\sin \theta_2> \\<br />
v_2 &= <|10|\cos \,5.640, |10|\sin \,5.640> \\<br />
v_2 &= <8, -6><br />
\end{aligned}

    EDIT: This long-winded way to solve the problem was deliberate.
    Last edited by eumyang; July 29th 2010 at 07:43 PM.
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  3. #3
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    pickslides's Avatar
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    You need to find the vector [x,y] where x^2+y^2= 10^2 and \frac{3}{-4}=\frac{x}{y}
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