Question

If $\vec{a}$ and $\vec{b}$ are unit vectors and $|\vec{a} + \vec{b}|=1$ then $|\vec{a} -\vec{b}|$ is equal to

• A. $\sqrt {2}$
• B. $1$
• C. $\sqrt {5}$
• D. $\sqrt {3}$

Answer 1

Answer: (D)

$\sqrt {3}$

Answer 2

The correct answer is D:$\sqrt{3}$
Given that;
$\vec{a},\vec{b}$ are unit vectors
$\therefore |a+b|=\sqrt{|\vec{a^2}|+|\vec{b^2}|+2|a||b|Cos\theta}=1$
⇒$|a||b|Cos\theta=\frac{1-1-1}{2}=-\frac{1}{2}$
Similarly;
$|\vec{a}-\vec{b}|=\sqrt{|\vec{a^2}|+|\vec{b^2}|-2|\vec{a}||\vec{b}|Cos\theta}$
$=\sqrt{(|\vec{a}|+|\vec{b}|)^2-4|a||b|Cos\theta}$
$=\sqrt{1^2-4|a||b|Cos\theta}$
$=\sqrt{1^2-4(-\frac{1}{2})}$
$=\sqrt{3}$