Let and be the vectors, and let be their common magnitude.
The resultant is times larger in magnitude than either vector alone, so .
Recall the dot product identity
where is the angle between the vectors and . In the special case of , we get
Now, to get the angle between and , we have
To compute the dot product, we take the dot product of the resultant with itself.
Solve for .
Since their dot product is zero, and are perpendicular, so .