Is it possible that the resultant of two vectors of equal magnitude has same magnitude as two vectors have and how?

Text Solution

Solution : Let the mangniture of each vector and of the resultant be a and the angle between the vectors be `alpha`. Hence, `a^2=a^2+a^2+2a*a cos alpha` or,` a^2=2a^2(1+cos alpha) or, 2(1+cos alpha)=1` or, `cos alpha =1/2 -1=-1/2= cos 120 ^@` `therefore alpha =120^@` Hence, the magnitude to the resultant of two equal vectors is equal to that of each of the given vectors when they are inclined at an angle of `120^@` with each other.

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.