Question

The sum of two vectors $\vec{a}$ and $\vec{b}$ is a vector $\vec{c}$, such that $ |\vec{a}|=|\vec{b}|=|\vec{c}|=2. $ Then, the magnitude of $ \vec{a} -\vec {b} $ is equal to

• A. $ \sqrt{3} $
• B. $ 2 $
• C. $ 2\sqrt{3} $
• D. $ 0 $

Answer 1

The Correct Option is C

Given, $ (a+b)=c $ Squaring on both sides, $ {{(a+b)}^{2}}={{c}^{2}} $ $ \Rightarrow $ $ (a+b)\,.\,(a+b)-|c{{|}^{2}} $ $ \Rightarrow $ $ |a{{|}^{2}}+|b{{|}^{2}}+2a.b=|c{{|}^{2}} $ $ \Rightarrow $ $ 4+4+2a.b=4 $ $ (\because \,\,|a|=|b|=|c|=2\,given) $ $ \Rightarrow $ $ a\,.\,b=-2 $ ..(i) Now, we have $ |a+b{{|}^{2}}=|a{{|}^{2}}+|b{{|}^{2}}-2a.b $ $ =4+4-2(-2) $ [from E (i)] $ \Rightarrow $ $ |a-b|=2\sqrt{3} $ $ \therefore $ Magnitude of $ (a-b)=2\sqrt{3} $