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Thread: Vectors p and q

  1. #1
    Member u2_wa's Avatar
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    Vectors p and q

    There are two coplanar vectors $\displaystyle p$ & $\displaystyle q$. If $\displaystyle ap+bq=0$ (scalar multiplication), what can be inferred about the values of a & b?

    In my view answer should be $\displaystyle a=b=0$ but how about if $\displaystyle p$ and $\displaystyle q$ are also parallel!
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  2. #2
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    Hello u2_wa
    Quote Originally Posted by u2_wa View Post
    There are two coplanar vectors $\displaystyle p$ & $\displaystyle q$. If $\displaystyle ap+bq=0$ (scalar multiplication), what can be inferred about the values of a & b?

    In my view answer should be $\displaystyle a=b=0$ but how about if $\displaystyle p$ and $\displaystyle q$ are also parallel!
    You are right: if $\displaystyle \vec p$ and $\displaystyle \vec q$ are non-zero, non-parallel vectors then $\displaystyle a = b = 0$.

    If $\displaystyle \vec p$ and $\displaystyle \vec q$ are parallel, then, for some $\displaystyle \lambda$:
    $\displaystyle \vec p =\lambda \vec q$

    i.e. $\displaystyle \vec p -\lambda\vec q = \vec 0$ (note: a zero vector)

    In which case, all you can say is that, for this value of $\displaystyle \lambda, \;a:b = -1:\lambda$.

    If $\displaystyle \vec p= \vec 0$, and $\displaystyle \vec q \ne \vec 0$, then $\displaystyle a\vec p + b\vec q = \vec 0 \Rightarrow b = 0$ and $\displaystyle a$ can take any value.

    Similarly if $\displaystyle \vec p\ne \vec 0$, and $\displaystyle \vec q = \vec 0$, then $\displaystyle a\vec p + b\vec q = \vec 0 \Rightarrow a = 0$ and $\displaystyle b$ can take any value.

    I think that just about covers all possible cases.

    Grandad
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