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Math Help - parallel vector problem

  1. #1
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    parallel vector problem

    given \vec{a}=\begin{pmatrix} h-2 \\ 5  \end{pmatrix} and \vec{b}=\begin{pmatrix} 6 \\ 2h  \end{pmatrix}.Find the value of h such that the vector \vec{a} is parallel to the vector \vec{b}
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  2. #2
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    If the vector \mathbf{a} is parallel to vector \mathbf{b}, then they have the same direction, and one will be the scalar multiple of another.

    Therefore

    \mathbf{a} = k\mathbf{b}

    \left[\begin{matrix}h-2\\5\end{matrix}\right] = k\left[\begin{matrix}6\\2h\end{matrix}\right]

    \left[\begin{matrix}h-2\\5\end{matrix}\right] = \left[\begin{matrix}6k\\2kh\end{matrix}\right].


    This means you end up with the system of equations

    h - 2 = 6k
    5 = 2kh

    Rearranging the second gives

    k = \frac{5}{2h}

    and substituting into the first gives

    h - 2 = 6\left(\frac{5}{2h}\right)

    h - 2 = \frac{15}{h}

    h^2 - 2h = 15

    h^2 - 2h - 15 = 0

    (h - 5)(h + 3) = 0

    h = 5 or h = -3.
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