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)

$2\sqrt{3}$

Answer 2

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 $(a-b)=2\sqrt{3}$