Find the resultant of the three vectors OA,OB and OC shown in figure .Radius of the circle is R . please give more explanation with solution

To solve this problem, we need to find the resultant of three vectors $\mathbf{OA}$, $\mathbf{OB}$, and $\mathbf{OC}$ that are given as position vectors on a circle of radius $R$.

### Step 1: Understanding the Problem
- The three vectors originate from the center of the circle and point towards points $A$, $B$, and $C$ on the circumference.
- The angle between the vectors is not explicitly given, but we assume they are symmetrically placed, meaning they form angles of ${120}^{\circ }$ with each other.

### Step 2: Representing Vectors in Component Form
Since the vectors are directed radially outward from the center and positioned symmetrically, we express them using unit vectors.

#### Vector Representation:
Let’s assume:
- $\mathbf{OA}$ lies along the positive x-axis.
- $\mathbf{OB}$ and $\mathbf{OC}$ make angles of ${120}^{\circ }$ and ${240}^{\circ }$ respectively with the x-axis.

Using unit vectors:
- $\mathbf{OA}=R\hat{i}$
- $\mathbf{OB}=R(\cos {120}^{\circ }\hat{i}+\sin {120}^{\circ }\hat{j})$
- $\mathbf{OC}=R(\cos {240}^{\circ }\hat{i}+\sin {240}^{\circ }\hat{j})$

Using trigonometric values:
- $\cos {120}^{\circ }=-\frac{1}{2}$, $\sin {120}^{\circ }=\frac{\sqrt{3}}{2}$
- $\cos {240}^{\circ }=-\frac{1}{2}$, $\sin {240}^{\circ }=-\frac{\sqrt{3}}{2}$

Thus:
- $\mathbf{OB}=R(-\frac{1}{2}\hat{i}+\frac{\sqrt{3}}{2}\hat{j})$
- $\mathbf{OC}=R(-\frac{1}{2}\hat{i}-\frac{\sqrt{3}}{2}\hat{j})$

### Step 3: Finding the Resultant
The resultant vector $\mathbf{R}$ is given by:
$$\mathbf{R}=\mathbf{OA}+\mathbf{OB}+\mathbf{OC}$$

#### x-component:
$$R_x=R+R(-\frac{1}{2})+R(-\frac{1}{2})$$
$$R_x=R-\frac{R}{2}-\frac{R}{2}=R-R=0$$

#### y-component:
$$R_y=0+R\left(\frac{\sqrt{3}}{2}\right)+R(-\frac{\sqrt{3}}{2})$$
$$R_y=R\frac{\sqrt{3}}{2}-R\frac{\sqrt{3}}{2}=0$$

### Step 4: Magnitude of Resultant
$$|\mathbf{R}|=\sqrt{R_x^2+R_y^2}=\sqrt{0^2+0^2}=0$$

### Conclusion:
The resultant of the three vectors is zero. This happens because the three vectors are symmetrically arranged around the center, leading to perfect cancellation of their components. Hence, the system is in equilibrium.

**Final Answer:** The resultant of the three vectors is **zero**.