Unit quaternion multiplied by -1 If all components of a unit quaternion (also known as versor) are multiplied by -1, so it still remains a versor, does the resulting versor is considered equivalent to the original versor?

If we think about rotation quaternion which are also unit quaternion then multiplying it with $-1$ will result it in additional $2\pi$ rotation and in consequence this will not affect original rotation hence it is equivalent to original unit quaternion. 
$$q = \cos{\frac{\theta}{2}}+\sin{\frac{\theta}{2}}\frac{\vec{u}}{\|\vec{u}||}$$$$-q= \cos{(\frac{\theta}{2}+\pi)}+\sin{(\frac{\theta}{2}+\pi)}\frac{\vec{u}}{\|\vec{u}||}$$
hence if $q$ is rotates through $\theta$ around $\vec{u}$ then $-q$ rotates $\theta +2\pi$. 
