Given that `vec(A)+vec(B)=vec(R)` and `vec(A)+2vec(B)` is perpendicular to `vec(A)` . Then :-

Options

A. 2B=R
B. B=2R
C. B=R
D. `B^(2)=2R^(2)`

Explanation

To solve the problem step by step, we will analyze the given information and apply vector properties.

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

### Step 2: Use the Perpendicular Condition
Since \( \vec{A} + 2\vec{B} \) is perpendicular to \( \vec{A} \), we can use the dot product:
\[ (\vec{A} + 2\vec{B}) \cdot \vec{A} = 0 \]
Expanding this gives:
\[ \vec{A} \cdot \vec{A} + 2\vec{B} \cdot \vec{A} = 0 \]
Let’s denote \( \vec{A} \cdot \vec{A} \) as \( A^2 \) (the magnitude squared of \( \vec{A} \)):
\[ A^2 + 2\vec{B} \cdot \vec{A} = 0 \]
From this, we can express \( \vec{B} \cdot \vec{A} \):
\[ 2\vec{B} \cdot \vec{A} = -A^2 \quad \Rightarrow \quad \vec{B} \cdot \vec{A} = -\frac{A^2}{2} \]

### Step 3: Substitute \( \vec{B} \) in Terms of \( \vec{R} \)
From the first equation, we can express \( \vec{B} \):
\[ \vec{B} = \vec{R} - \vec{A} \]
Now, substitute \( \vec{B} \) into the expression for \( \vec{B} \cdot \vec{A} \):
\[ (\vec{R} - \vec{A}) \cdot \vec{A} = -\frac{A^2}{2} \]
Expanding this gives:
\[ \vec{R} \cdot \vec{A} - \vec{A} \cdot \vec{A} = -\frac{A^2}{2} \]
Substituting \( \vec{A} \cdot \vec{A} = A^2 \):
\[ \vec{R} \cdot \vec{A} - A^2 = -\frac{A^2}{2} \]
Rearranging gives:
\[ \vec{R} \cdot \vec{A} = \frac{A^2}{2} \]

### Step 4: Find the Magnitude of \( \vec{R} \)
Now we will find the magnitude of \( \vec{R} \):
Using the equation \( \vec{R} = \vec{A} + \vec{B} \) and substituting \( \vec{B} \):
\[ \vec{R} = \vec{A} + (\vec{R} - \vec{A}) = \vec{R} \]
To find \( R^2 \):
\[ R^2 = A^2 + B^2 + 2\vec{A} \cdot \vec{B} \]
Substituting \( \vec{B} = \vec{R} - \vec{A} \):
\[ R^2 = A^2 + (\vec{R} - \vec{A})^2 + 2\vec{A} \cdot (\vec{R} - \vec{A}) \]
This leads to:
\[ R^2 = A^2 + B^2 + 2\left(\frac{A^2}{2}\right) = A^2 + B^2 + A^2 = 2A^2 + B^2 \]

### Step 5: Conclusion
From the previous steps, we can conclude that the relationship between the vectors leads us to find that \( \vec{B} = \frac{1}{2}\vec{R} \).

### Final Answer
Thus, the correct option is that \( \vec{B} = \frac{1}{2}\vec{R} \).

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