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Thread: find a unit vector and a vector of magnitude 3 in the opposide direction

  1. #1
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    find a unit vector and a vector of magnitude 3 in the opposide direction

    seeing:
    A(4,0) B(2,1)

    find a unit vector in the direction of AB ( smal arrow above AB that begins from left to right) abd a vector of magnitude 3 in the opposite direction

    thank you
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  2. #2
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    A unit vector is simply a vector with a magnitude of 1. To get the unit vector in the direction of the vector $\displaystyle \overrightarrow{A B}$, all you have to do is divide the components of the vector by its magnitude. This process is called normalization.

    $\displaystyle \vec{AB}$
    $\displaystyle = (2 - 4, 1 - 0)$
    $\displaystyle = (-2, 1)$

    Magnitude of $\displaystyle \vec{AB}$
    $\displaystyle = \sqrt {{x}^{2}+{y}^{2}}$
    $\displaystyle = \sqrt {{(-2)}^{2}+{1}^{2}}$
    $\displaystyle = \sqrt {5}$

    Unit vector in the direction of $\displaystyle \vec{AB}$
    $\displaystyle = \left ( \frac{-2}{\sqrt{5}}, \frac{1}{\sqrt{5}} \right )$

    To find the vector of magnitude 3 in the opposite direction of the vector $\displaystyle \vec{AB}$, you have to find the unit vector in the direction of $\displaystyle \vec{BA}$, which is in the opposite direction. Since the unit vector has a magnitude of 1, and you need a vector of magnitude 3, just multiply the unit vector by 3, and you'll get the answer.
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  3. #3
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    thank u very much!
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