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,
${(a+b)}^2=c^2$
$\Rightarrow$ $(a+b).(a+b)-|c{|}^2$
$\Rightarrow$ $|a{|}^2+|b{|}^2+2a.b=|c{|}^2$
$\Rightarrow$ $4+4+2a.b=4$
$(\because |a|=|b|=|c|=2given)$
$\Rightarrow$ $a.b=-2$ ..(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 $(a-b)=2\sqrt{3}$