A

Step 1: Given three vectors: \( A = R \hat{i} \), \( B = R \hat{j} \), and \( C = R \hat{i} e^{i45^{\circ}} \).
Step 2: Finding vector \( C \):
\( C = R ( \cos 45^{\circ} \hat{i} + \sin 45^{\circ} \hat{j}) = R \left( \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \right) \)
Step 3: Add the vectors together:
$$ R_{total} = (R + R \cdot \frac{1}{\sqrt{2}})\hat{i} + (R + R \cdot \frac{1}{\sqrt{2}})\hat{j} $$
Step 4: Calculate the total force:
Resultant vector = \( R_{total} = R(1 + \frac{1}{\sqrt{2}})\hat{i} + R(1 + \frac{1}{\sqrt{2}})\hat{j} \).
The magnitude will be the same in both directions, and is expressed as:
$$ |R_{total}| = R \sqrt{(1 + \frac{1}{\sqrt{2}})^2 + (1 + \frac{1}{\sqrt{2}})^2} = 2R \left(1 + \frac{1}{\sqrt{2}}\right) $$
Therefore, the resultant vector of these three vectors can be expressed in terms of radius R as Option A.