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Thread: 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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    $\displaystyle v_1 = <-4, 3>$

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

    $\displaystyle \theta_2 = \theta_1 + \pi \approx 2.498 + \pi \approx 5.640 \,rad$
    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 $\displaystyle [0, 2\pi)$.

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

    EDIT: This long-winded way to solve the problem was deliberate.
    Last edited by eumyang; Jul 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 $\displaystyle [x,y]$ where $\displaystyle x^2+y^2= 10^2$ and $\displaystyle \frac{3}{-4}=\frac{x}{y}$
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