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# The greatest value of $|\overrightarrow a+\overrightarrow b|+|\overrightarrow a-\overrightarrow b|$ where $\overrightarrow a\:\:and\:\:\overrightarrow b$ are unit vectors is ?

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• Max. value of $acos\theta+bsin\theta= \sqrt {a^2+b^2}$
$|\overrightarrow a+\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+2|\overrightarrow a|\overrightarrow b|cos\theta$
$=2(1+cos\theta)=4cos^2\large\frac{\theta}{2}$
$\Rightarrow\:|\overrightarrow a+\overrightarrow b|=2cos\large\frac{\theta}{2}$
Similarly $|\overrightarrow a-\overrightarrow b|^2=4sin^2\large\frac{\theta}{2}$
$\Rightarrow\:|\overrightarrow a-\overrightarrow b|=2sin\large\frac{\theta}{2}$
$\therefore\:$ Max. value of $|\overrightarrow a+\overrightarrow b| +|\overrightarrow a-\overrightarrow b|$ is
Max. Value of $2(cos\large\frac{\theta}{2}$$+sin\large\frac{\theta}{2})$
$=2\sqrt 2$