If\(\left| {\overrightarrow a } \right|\) = \(\left| {\overrightarrow b } \right|\) = \(\left| {\overrightarrow c } \right|\) and  \(\overrightarrow a \, + \,\overrightarrow b \, = \,\,\overrightarrow c \), then angle between \(\overrightarrow a \) and \(\overrightarrow b \) is -

We Have,

If, \(\left| {\overrightarrow a } \right|\) = \(\left| {\overrightarrow b } \right|\) = \(\left| {\overrightarrow c } \right|\)

⇒ a = b = c .... (1)

And,

\(\overrightarrow a \, + \,\overrightarrow b \, = \,\,\overrightarrow c \) .... (2)

Using the concept of, Parallelogram Law of Vector Addition: It is used to add two vector quantities.

\(\overrightarrow A \, + \,\overrightarrow B \, = \,\,\overrightarrow R\)

R2 = A2 + B2 +2AB cos θ 

From equation (2),

c2 = a2 + b2 +2ab cos θ

Since,

a = b = c

Hence,

a² = a² + a² +2a² cos θ

or, a2 = a2 (1 + 1 +2cos θ)

or, 1 = 2 + 2 cos θ

or, 2 cos θ = -1

or, cos θ = (-0.5)

Hence, θ = cos^-1 (0.5) = 120° = \(\frac{{2\pi }}{3}\)