Resultant of two vector of equal magnitude A is

Options

A. `sqrt(3)Aat 60^(@)`
B. `sqrt(2)Aat 90^(@)`
C. `2Aat 120^(@)`
D. `Aat 180^(@)`

Explanation

To find the resultant of two vectors of equal magnitude \( A \), we can follow these steps:

### Step 1: Understand the Problem
We have two vectors, both with the same magnitude \( A \). We need to find the resultant vector based on the angle \( \theta \) between them.

### Step 2: Use the Resultant Formula
The formula for the resultant \( R \) of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
Since both vectors have equal magnitudes, we can replace \( B \) with \( A \):
\[ R = \sqrt{A^2 + A^2 + 2A \cdot A \cos \theta} \]

### Step 3: Simplify the Expression
Now, simplify the expression:
\[ R = \sqrt{2A^2 + 2A^2 \cos \theta} \]
\[ R = \sqrt{2A^2(1 + \cos \theta)} \]
\[ R = A\sqrt{2(1 + \cos \theta)} \]

### Step 4: Use Trigonometric Identity
Using the trigonometric identity \( 1 + \cos \theta = 2 \cos^2(\theta/2) \):
\[ R = A\sqrt{2 \cdot 2 \cos^2(\theta/2)} \]
\[ R = A \cdot 2 \cos(\theta/2) \]
\[ R = 2A \cos(\theta/2) \]

### Step 5: Evaluate for Different Angles
Now we can evaluate the resultant for different angles \( \theta \):

1. **For \( \theta = 60^\circ \)**:
\[ R = 2A \cos(30^\circ) = 2A \cdot \frac{\sqrt{3}}{2} = \sqrt{3}A \]

2. **For \( \theta = 90^\circ \)**:
\[ R = 2A \cos(45^\circ) = 2A \cdot \frac{1}{\sqrt{2}} = \sqrt{2}A \]

3. **For \( \theta = 120^\circ \)**:
\[ R = 2A \cos(60^\circ) = 2A \cdot \frac{1}{2} = A \]

4. **For \( \theta = 180^\circ \)**:
\[ R = 2A \cos(90^\circ) = 2A \cdot 0 = 0 \]

### Conclusion
From the above calculations, we find:
- For \( \theta = 60^\circ \), \( R = \sqrt{3}A \)
- For \( \theta = 90^\circ \), \( R = \sqrt{2}A \)
- For \( \theta = 120^\circ \), \( R = A \)
- For \( \theta = 180^\circ \), \( R = 0 \)

Thus, the correct options based on the resultant values are those corresponding to \( \sqrt{3}A \) and \( \sqrt{2}A \).