To satisfy the last three equations, either or , , and are all 0. The latter is impossible because a is a real number and the first equation would imply that Therefore, and In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere.
To satisfy the last three equations, eith:er or , , and are all 0. The latter is impossible because a is a real number and the first equation would imply that Therefore, and In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit !sphere.