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Math Help - Urgent: Vectors

  1. #1
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    Urgent: Vectors

    Find the value of parameter m that vector p->=ma->+17b-> and vector q->=3a->-b-> would be across if |a->|=2 and |b->|5 and the angle between a-> and b-> is 120 degree.


    The math book shows that the answer is 40.
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  2. #2
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    Re: Urgent: Vectors

    \vec{p}=m\vec{a}+17\vec{b}
    \vec{q}=3\vec{a}-\vec{b}

    \vec{p}\bot \vec{q}\Rightarrow \vec{p}\cdot \vec{q}=0
    \begin{align}<br />
\left( m\vec{a}+17\vec{b} \right)\cdot \left( 3\vec{a}-\vec{b} \right)&=0 \\ <br />
3ma\cdot a-ma\cdot b+51a\cdot b-17b\cdot b&=0 \\ <br />
3m|a{{|}^{2}}+\left( 51-m \right)a\cdot b-17|b{{|}^{2}}&=0 \\ <br />
3m|a{{|}^{2}}+\left( 51-m \right)|a||b|\cos 120{}^\circ -17|b{{|}^{2}}&=0 \\ <br />
3m{{\left( 2 \right)}^{2}}+\left( 51-m \right)\left( 2 \right)\left( 5 \right)\left( -\frac{1}{2} \right)-17{{\left( 5 \right)}^{2}}&=0 \\ <br />
12m-255+5m-425&=0 \\ <br />
17m&=680 \\ <br />
m&=40  <br />
\end{align}
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