# Find Value of |(vec(a) xx vec(c)) * vec(b)|

If $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are two unit vectors and $\stackrel{\to }{c}$ be a vector such that $2\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)+\stackrel{\to }{c}=\stackrel{\to }{b}×\stackrel{\to }{c}$. Then maximum value of $|\left(\stackrel{\to }{a}×\stackrel{\to }{c}\right)\cdot \stackrel{\to }{b}|$
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Cody Petty
Let $\stackrel{\to }{c}=\alpha \stackrel{\to }{a}+\beta \stackrel{\to }{b}+\gamma \stackrel{\to }{a}×\stackrel{\to }{b}$
$\begin{array}{rl}\stackrel{\to }{b}×\stackrel{\to }{c}& =\alpha \stackrel{\to }{b}×\stackrel{\to }{a}+\beta \stackrel{\to }{b}×\stackrel{\to }{b}+\gamma \stackrel{\to }{b}×\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)\\ & =-\alpha \stackrel{\to }{a}×\stackrel{\to }{b}+\gamma \left[\left(\stackrel{\to }{b}\cdot \stackrel{\to }{b}\right)\stackrel{\to }{a}-\left(\stackrel{\to }{b}\cdot \stackrel{\to }{a}\right)\stackrel{\to }{b}\right]\\ & =\gamma \stackrel{\to }{a}-\gamma \left(\stackrel{\to }{a}\cdot \stackrel{\to }{b}\right)\stackrel{\to }{b}-\alpha \stackrel{\to }{a}×\stackrel{\to }{b}\end{array}$
$2\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)+\stackrel{\to }{c}=\alpha \stackrel{\to }{a}+\beta \stackrel{\to }{b}+\left(\gamma +2\right)\stackrel{\to }{a}×\stackrel{\to }{b}$
Therefore, $\gamma =\alpha$, $-\gamma \left(\stackrel{\to }{a}\cdot \stackrel{\to }{b}\right)=\beta$ and $-\alpha =\gamma +2$
Solving, $\alpha =\gamma =-1$ and $\beta =\stackrel{\to }{a}\cdot \stackrel{\to }{b}$
So, $\stackrel{\to }{c}=-\stackrel{\to }{a}+\left(\stackrel{\to }{a}\cdot \stackrel{\to }{b}\right)\stackrel{\to }{b}-\stackrel{\to }{a}×\stackrel{\to }{b}$
$\begin{array}{rl}\stackrel{\to }{c}\cdot \stackrel{\to }{c}& =\stackrel{\to }{a}\cdot \stackrel{\to }{a}+\left(\stackrel{\to }{a}\cdot \stackrel{\to }{b}{\right)}^{2}\left(\stackrel{\to }{b}\cdot \stackrel{\to }{b}\right)+\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)\cdot \left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)-2\left(\stackrel{\to }{a}\cdot \stackrel{\to }{b}\right)\left(\stackrel{\to }{a}\cdot \stackrel{\to }{b}\right)\\ |\stackrel{\to }{c}{|}^{2}& =1-\left(\stackrel{\to }{a}\cdot \stackrel{\to }{b}{\right)}^{2}+|\stackrel{\to }{a}×\stackrel{\to }{b}{|}^{2}\\ & =2|\stackrel{\to }{a}×\stackrel{\to }{b}{|}^{2}\end{array}$
and its greatest value is 2.
The greatest value of $|\left(\stackrel{\to }{a}×\stackrel{\to }{c}\right)\cdot \stackrel{\to }{b}|$ is 1.