If `vecA + vecB = vecR` and `2vecA + vecB` s perpendicular to `vecB` then

Options

A. A = R
B. B = 2R
C. B = R
D. B=A

Answer

To solve the problem step by step, we will analyze the given vectors and the conditions provided.

### Step 1: Understand the Given Information
We are given two vectors:
1. \( \vec{A} + \vec{B} = \vec{R} \)
2. \( 2\vec{A} + \vec{B} \) is perpendicular to \( \vec{B} \)

### Step 2: Use the Condition of Perpendicularity
Since \( 2\vec{A} + \vec{B} \) is perpendicular to \( \vec{B} \), we can use the dot product to express this condition:
\[ (2\vec{A} + \vec{B}) \cdot \vec{B} = 0 \]

### Step 3: Expand the Dot Product
Expanding the dot product gives:
\[ 2\vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{B} = 0 \]
This simplifies to:
\[ 2\vec{A} \cdot \vec{B} + |\vec{B}|^2 = 0 \]

### Step 4: Rearrange the Equation
Rearranging the equation gives:
\[ 2\vec{A} \cdot \vec{B} = -|\vec{B}|^2 \]
Dividing both sides by 2:
\[ \vec{A} \cdot \vec{B} = -\frac{|\vec{B}|^2}{2} \]

### Step 5: Substitute into the Magnitude Equation
Now, we know that:
\[ |\vec{A} + \vec{B}| = |\vec{R}| \]
Taking the magnitude of both sides:
\[ |\vec{A} + \vec{B}| = |\vec{R}| \]
Using the formula for the magnitude of the sum of two vectors:
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2\vec{A} \cdot \vec{B}} \]

### Step 6: Substitute \( \vec{A} \cdot \vec{B} \)
Substituting \( \vec{A} \cdot \vec{B} = -\frac{|\vec{B}|^2}{2} \) into the equation:
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2\left(-\frac{|\vec{B}|^2}{2}\right)} \]
This simplifies to:
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 - |\vec{B}|^2} \]
Thus:
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2} \]
Which means:
\[ |\vec{A} + \vec{B}| = |\vec{A}| \]

### Step 7: Relate to \( |\vec{R}| \)
Since \( |\vec{A} + \vec{B}| = |\vec{R}| \), we have:
\[ |\vec{R}| = |\vec{A}| \]
Thus, we conclude:
\[ |\vec{A}| = |\vec{R}| \]

### Final Result
This implies that the magnitude of vector \( \vec{A} \) is equal to the magnitude of vector \( \vec{R} \).

### Conclusion
Thus, the final relation is:
\[ |\vec{A}| = |\vec{R}| \]