Let $\vec{a}, \vec{b}$ be two unit vectors whose sum is also a unit vector $\vec{c}$

$\vec{a}+\vec{b}=\vec{c} \ldots \ldots .$. (i) and $|\vec{a}|=|\vec{b}|=|\vec{c}|=1$

Now, magnitude of sum of $\vec{a}$ and $\vec{b}$ is

$|\vec{a}+\vec{b}|=|\vec{a}|^2+|\vec{b}|^2+2(\vec{a} \cdot \vec{b})$

$\Rightarrow 1=1+1+2(\vec{a} \cdot \vec{b}) \ldots .$. From $(i)$

$\Rightarrow \vec{a} \cdot \vec{b}=-\frac{1}{2}$

Magnitude of their difference is given by

$|\vec{a}-\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2-2 \vec{a} \cdot \vec{b}$

$=1+1+2 \times \frac{1}{2}=3$

$\Rightarrow|a-b|=\sqrt{3}$