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

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Let the given unit vectors be $\overrightarrow a\:\:and\:\: \overrightarrow b$
Given: $|\overrightarrow a+\overrightarrow b|=1=|\overrightarrow a|=|\overrightarrow b|$
Now
$|\overrightarrow a +\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+2\overrightarrow a.\overrightarrow b$
$\Rightarrow\:1=1+1+2\overrightarrow a.\overrightarrow b$
$\Rightarrow\:\overrightarrow a.\overrightarrow b=-\large\frac{1}{2}$
and
$|\overrightarrow a -\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2-2\overrightarrow a.\overrightarrow b$
$=1+1+1=3$
$\therefore\:|\overrightarrow a-\overrightarrow b|=\sqrt 3$
answered Dec 3, 2013