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Math Help - Vector problem

  1. #1
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    Vector problem

    Givan a=3x+4y and b =(h+5)x-(2k-3)y,find in term of h,k,x and y,vector a-2b.If the vector a-2b is a zero vector and x and y are non-zero vectors,find the values of h and k.
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  2. #2
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    Hello, mastermin346!

    This is quite straight-forward.
    Exactly where is your difficulty?


    Given: . \begin{array}{ccc}\vec a &= & 3\vec x+4\vec y \\ \vec b &=& (h+5)\vec x -(2k-3)\vec y\end{array}

    . . find \vec a-2\vec b in terms of h,k, \vec x,\vec y.

    \vec a - 2\vec b \;=\;(3\vec x-4\vec y) - 2\bigg[(h+5)\vec x - (2k-3)\vec y\bigg]

    . . . . =\;3\vec x+ 4\vec y - 2(h+5)\vec x + 2(2k-3)\vec y

    . . . . =\; 3\vec x + 4\vec y - 2h\vec x - 10\vec x + 4k\vec y - 6\vec y

    . . . . =\;-2h\vec x - 7\vec x + 4k\vec y - 2\vec y

    . . . . =\;-(2h+7)\vec x + 2(2k-1)\vec y




    If the vector a -2b is the zero vector and x and y are non-zero vectors,
    . . find the values of h and k.

    If \vec a - 2\vec b \:=\:\vec0, then: . \begin{array}{ccccccccc}\text{-}(2h+7) \;=\;0 & \Rightarrow & h \;=\;\text{-}\frac{7}{2} \\ \\[-3mm] 2(2k-1) \;=\;0 & \Rightarrow & k \;=\; \frac{1}{2} \end{array}

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