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Vectors a,b |a+kb|=1 .Show that |a||b|sinx<=b, x:angle of vectors a,b

  1. Oct 19, 2011 #1
    1. The problem statement, all variables and given/known data

    Let there be two vectors [tex]\mathbf{OA},\mathbf{OB}\neq\mathbf{0}[/tex]If [tex]
    \exists k\in \mathbb{R}[/tex] such as that [tex]\left \| \mathbf{OA} +k\mathbf{OB}\right \|=1[/tex] show that [tex]Area(OACB)\leq\left \| \mathbf{OB} \right \|[/tex] (OACB:parallelogram)

    2. Relevant equations

    3. The attempt at a solution

    I proved that we need to show that [tex]\left \|\mathbf{a}\right \| \left \|\mathbf{b}\right \| \sin(\theta )\leq \left \|\mathbf{b} \right \|[/tex] where θ:angle of vectors a=ΟΑ,b=ΟΒ but after that I am stuck.
    Any suggestions? Any hints on how I should proceed?
    Last edited: Oct 19, 2011
  2. jcsd
  3. Oct 19, 2011 #2
    Nevermind, I solved it. Here is the solution

    First of all,
    [tex]\left \| \mathbf{a} +k\mathbf{b}\right \|=1\Leftrightarrow (\mathbf{a} +k\mathbf{b})^{2}=1\Leftrightarrow \mathbf{a}^{2} +k^{2}\mathbf{b}^{2}+2k\left \langle {\mathbf{a} ,\mathbf{b} } \right\rangle =1\Leftrightarrow\left \langle {\mathbf{a} ,\mathbf{b} } \right\rangle^{2}=\frac{(1-\mathbf{a}^{2} -k^{2}\mathbf{b}^{2} )^{2}}{4k^2} (1) [/tex]

    We need to show that
    [tex]\left \|\mathbf{a}\right \| \left \|\mathbf{b}\right \| \sin(\theta )\leq \left \|\mathbf{b} \right \|\Leftrightarrow \left \|\mathbf{a}\right \|^{2} \left \|\mathbf{b}\right \|^{2} \sin(\theta )^{2}\leq \left \|\mathbf{b} \right \|^{2}\Leftrightarrow \left \|\mathbf{a}\right \|^{2} \left \|\mathbf{b}\right \|^{2}- \left \|\mathbf{a}\right \|^{2} \left \|\mathbf{b}\right \|^{2}\cos(\theta )^{2}\leq\left \|\mathbf{b} \right \|^{2}\
    \Leftrightarrow\left \|\mathbf{a}\right \|^{2} \left \|\mathbf{b}\right \|^{2}-\left \|\mathbf{b} \right \|^{2}\leq\left\langle {\mathbf{a} ,\mathbf{b} } \right\rangle^{2}(2)[/tex]


    (2)\overset{(1)}{\rightarrow}\mathbf{a}^{2} \mathbf{b}^{2}-\mathbf{b} ^{2}\leq\frac{(1-\mathbf{a}^{2}-k^{2}\mathbf{b}^{2}) ^{2}}{4k^2}\Leftrightarrow (1-\mathbf{a}^{2}+k^{2}\mathbf{b}^{2})^{2}\geq 0

    which is true!
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