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Thread: Vectors

  1. #1
    Member Mr Rayon's Avatar
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    Vectors

    Consider the following relationships between vectors $\displaystyle u$, $\displaystyle v$ and $\displaystyle w$.

    i) $\displaystyle u = 2v + w$
    ii) $\displaystyle w = v - u$

    Which of the following statements is true?


    A) $\displaystyle u = w$
    B) $\displaystyle u = v$

    C) $\displaystyle u = \frac{2}{3}v$

    D) $\displaystyle u = \frac{3}{2}v$

    E) $\displaystyle u = 3v$


    Please explain why to make me understand.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Mr Rayon View Post
    Consider the following relationships between vectors $\displaystyle u$, $\displaystyle v$ and $\displaystyle w$.

    i) $\displaystyle u = 2v + w$
    ii) $\displaystyle w = v - u$

    Which of the following statements is true?


    A) $\displaystyle u = w$

    B) $\displaystyle u = v$

    C) $\displaystyle u = \frac{2}{3}v$

    D) $\displaystyle u = \frac{3}{2}v$

    E) $\displaystyle u = 3v$


    Please explain why to make me understand.
    Note that substituting (ii) into (i) gives us $\displaystyle u=2v+v-u\implies 2u=3v\implies u=\tfrac{3}{2}v$ => (D) is true. Thus, its clear that (B), (C), and (E) are false.

    Note that subsituting (i) into (ii) gives us $\displaystyle w=v-\left(2v+w\right)\implies w=-v-w\implies 2w=-v\implies 2w=-\tfrac{2}{3}u\implies 3w=-u$, which implies (A) is false.

    Thus, (D) is the only one that is true.
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