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# If the sum of two unit vectors is a unit vector, then the magnitude of their difference is ?

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Let the two unit vectors be $\overrightarrow a$ and $\overrightarrow b$
Given $|\overrightarrow a+\overrightarrow b|=1$
$\Rightarrow\:|\overrightarrow a+\overrightarrow b|^2=1$
$\Rightarrow\:|\overrightarrow a|^2+|\overrightarrow b|^2+2\overrightarrow a.\overrightarrow b=1$
$\Rightarrow\:2\overrightarrow a.\overrightarrow b=-1$
$|\overrightarrow a-\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2-2\overrightarrow a.\overrightarrow b$
$=2+1=3$
$\therefore\:|\overrightarrow a-\overrightarrow b|=\sqrt 3$
answered Nov 30, 2013