Resultant of two vectors one having magnitude twice of other is perpendicular to the smaller vector then find the angle `( " in"^@)` between the vectors ?

Explanation

To solve the problem, we need to find the angle between two vectors, where one vector (let's call it vector A) has a magnitude that is half of the other vector (vector B), and the resultant of these two vectors is perpendicular to the smaller vector (vector A).

### Step-by-step Solution:

1. **Define the Vectors**:
- Let the magnitude of vector A be \( A \).
- Since vector B has twice the magnitude of vector A, we have \( B = 2A \).

2. **Understanding the Resultant**:
- The resultant vector \( R \) of vectors A and B is given by the vector addition formula. Since the resultant is perpendicular to vector A, we can use the properties of right triangles.

3. **Using the Cosine Rule**:
- According to the cosine rule in a triangle formed by the vectors, we have:
\[ R^2 = A^2 + B^2 + 2AB \cos(\theta) \]
- Here, \( R \) is the resultant, \( \theta \) is the angle between vectors A and B, and \( B = 2A \).

4. **Substituting Values**:
- Substitute \( B = 2A \) into the equation:
\[ R^2 = A^2 + (2A)^2 + 2A(2A) \cos(\theta) \]
\[ R^2 = A^2 + 4A^2 + 4A^2 \cos(\theta) \]
\[ R^2 = 5A^2 + 4A^2 \cos(\theta) \]

5. **Using the Perpendicular Condition**:
- Since the resultant \( R \) is perpendicular to vector A, we know that \( R \cdot A = 0 \). This means:
\[ R^2 = B^2 - A^2 \]
- Substitute \( B = 2A \):
\[ R^2 = (2A)^2 - A^2 = 4A^2 - A^2 = 3A^2 \]

6. **Equating the Two Expressions for R²**:
- From the previous steps, we have two expressions for \( R^2 \):
\[ 3A^2 = 5A^2 + 4A^2 \cos(\theta) \]
- Rearranging gives:
\[ 3A^2 - 5A^2 = 4A^2 \cos(\theta) \]
\[ -2A^2 = 4A^2 \cos(\theta) \]
- Dividing both sides by \( 4A^2 \):
\[ \cos(\theta) = -\frac{1}{2} \]

7. **Finding the Angle**:
- The angle \( \theta \) that satisfies \( \cos(\theta) = -\frac{1}{2} \) is:
\[ \theta = 120^\circ \quad \text{(or } \theta = 240^\circ \text{, but we consider the acute angle here)} \]

### Final Answer:
The angle between vector A and vector B is \( 120^\circ \).