For which angle between two equal vectors will the magnitude of the sum of two vectors be equal to the magnitude of each vector?

If the magnitude of the resultant of two vectors of equal magnitudes is equal to the magnitude of either vectors, then the angle between two vectors is

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The question asks whether it is possible to add two 'equal' vectors, and end up with a vector whose magnitude is equal to the magnitudes of both the vectors. The wording is not very clear: either the two vectors are themselves equal, or their magnitudes are equal.

If you add two vectors with equal magnitude, and the magnitude of the resultant vector is equal to the magnitude of both vectors, then the three vectors obviously form an equilateral triangle. As @almagest said, this means that the difference between the angles of the two vectors is $120$ degrees.

If the vectors are equal, then their sum will necessarily have a larger magnitude than either of them unless the vector is zero.