The resultant of two vector A and B is at right angles to A and its magnitude is half of B. Find the angle between A and B.

Explanation

To solve the problem, we need to find the angle between two vectors A and B, given that their resultant is perpendicular to vector A and has a magnitude that is half of vector B.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two vectors A and B. The resultant vector R = A + B is at a right angle (90 degrees) to vector A. We also know that the magnitude of the resultant vector R is half the magnitude of vector B.

2. **Setting Up the Vectors**:
Let's denote the angle between vectors A and B as θ. We can use the cosine and sine components of vector B with respect to vector A:
- The horizontal component of B (along A) is \( B \cos(\theta) \).
- The vertical component of B (perpendicular to A) is \( B \sin(\theta) \).

3. **Finding the Resultant**:
Since the resultant R is perpendicular to A, we can express the resultant in terms of its components:
- The component of R along A is \( A + B \cos(\theta) \) (which must equal 0 since R is perpendicular to A).
- The vertical component of R is \( B \sin(\theta) \).

4. **Setting Up the Equations**:
From the above, we have:
\[ A + B \cos(\theta) = 0 \quad \text{(1)} \]
This implies:
\[ A = -B \cos(\theta) \quad \text{(2)} \]

5. **Magnitude of the Resultant**:
The magnitude of the resultant R is given as:
\[ |R| = \sqrt{(A + B \cos(\theta))^2 + (B \sin(\theta))^2} \]
Since \( |R| = \frac{1}{2} |B| \), we can write:
\[ |R| = \sqrt{0 + (B \sin(\theta))^2} = B \sin(\theta) \]
Therefore, we have:
\[ B \sin(\theta) = \frac{1}{2} B \]
This simplifies to:
\[ \sin(\theta) = \frac{1}{2} \quad \text{(3)} \]

6. **Finding the Angle θ**:
From equation (3), we know that:
\[ \theta = 30^\circ \quad \text{or} \quad \theta = 150^\circ \]

7. **Finding the Angle Between A and B**:
Since we are looking for the angle between A and B, and we know that the angle between A and the resultant is 90 degrees, we can find the angle between A and B:
\[ \text{Angle between A and B} = 90^\circ + \theta \]
Thus:
\[ \text{Angle between A and B} = 90^\circ + 60^\circ = 150^\circ \]

### Final Answer:
The angle between vectors A and B is **150 degrees**.