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

Answer: (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 Eq. (i)]
$ \Rightarrow $ $ |a-b|=2\sqrt{3} $
$ \therefore $ Magnitude of $ (a-b)=2\sqrt{3} $