Four vectors `bara ,barb,barc, and bard` are lying in the same plane. The vectors `bara` and `barb` are equal in magnitude and inclined to each other at an angle of `120^@`.`barC` is the resultant of `bara and barb`. Further `bara + barb + bard=0.` If the angle between `bara and bard `is `beta ` and the angle between `bara and barc `is a, find the correct relation between `alpha ` and `beta`

Explanation

To solve the problem, we will analyze the vectors step by step and derive the relationship between the angles \( \alpha \) and \( \beta \).

### Step 1: Understand the Vectors
We have four vectors \( \vec{a}, \vec{b}, \vec{c}, \) and \( \vec{d} \) lying in the same plane. The vectors \( \vec{a} \) and \( \vec{b} \) are equal in magnitude and inclined to each other at an angle of \( 120^\circ \).

### Step 2: Resultant of Vectors \( \vec{a} \) and \( \vec{b} \)
Since \( \vec{c} \) is the resultant of \( \vec{a} \) and \( \vec{b} \), we can use the law of cosines to find the magnitude of \( \vec{c} \):
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