Given information:
- We have two vectors A and B.
- The resultant of these two vectors is perpendicular to vector A.
- The magnitude of the resultant is equal to half the magnitude of vector B.

To find:
The angle between vectors A and B.

Solution:

1. Representing the vectors:
Let's represent vector A as A and vector B as B.

2. Understanding the resultant vector:
The resultant vector is perpendicular to vector A. This means that the dot product of the resultant vector and vector A is zero. Mathematically, this can be represented as:

A · (A + B) = 0

Expanding the dot product:

A · A + A · B = 0

Since A · A is the magnitude of vector A squared (|A|^2), we can rewrite the equation as:

|A|^2 + A · B = 0

3. Magnitude of the resultant vector:
The magnitude of the resultant vector is equal to half the magnitude of vector B. Mathematically, this can be represented as:

|A + B| = (1/2) |B|

Squaring both sides of the equation:

(A + B) · (A + B) = (1/4) B^2

Expanding the dot product:

A · A + 2A · B + B · B = (1/4) B^2

Since A · A is the magnitude of vector A squared (|A|^2) and B · B is the magnitude of vector B squared (|B|^2), we can rewrite the equation as:

|A|^2 + 2A · B + |B|^2 = (1/4) |B|^2

4. Solving the equations:
We have two equations obtained from step 2 and step 3. Let's solve these equations simultaneously.

From equation 2: |A|^2 + A · B = 0
From equation 3: |A|^2 + 2A · B + |B|^2 = (1/4) |B|^2

Substituting the value of |A|^2 + A · B from equation 2 into equation 3:

0 + 2A · B + |B|^2 = (1/4) |B|^2

Simplifying the equation:

2A · B = - (3/4) |B|^2

Dividing both sides of the equation by |B|^2:

(2A · B) / |B|^2 = - (3/4)

The left side of the equation is the cosine of the angle between vectors A and B, so we can rewrite the equation as:

cosθ = - (3/4)

Taking the inverse cosine (cos^-1) of both sides of the equation:

θ = cos^-1(- (3/4))

5. Evaluating the angle:
Using a calculator, we can find the inverse cosine of - (3/4) to evaluate the angle θ.

θ ≈ 135.26 degrees

6. Answer:
The angle between vectors A and B is approximately 135.26 degrees.

Conclusion:
The angle between vectors A and B is approximately