Angular between two vectors `vec A and vecB` is `theta` . Resulatnt of the two makes an angle `theta//2` with the vector `vecA`. Select the correct option .

Options

A. `A=1//B`
B. `A=2B`
C. `A=B`
D. None of these

Explanation

To solve the problem, we need to analyze the relationship between the two vectors \(\vec{A}\) and \(\vec{B}\) and their resultant vector. Let's break it down step-by-step.

### Step 1: Understand the Given Information
We have two vectors, \(\vec{A}\) and \(\vec{B}\), which form an angle \(\theta\) between them. The resultant vector \(\vec{R}\) of these two vectors makes an angle of \(\frac{\theta}{2}\) with vector \(\vec{A}\).

### Step 2: Set Up the Diagram
Draw the vectors \(\vec{A}\) and \(\vec{B}\) such that they form an angle \(\theta\). The resultant vector \(\vec{R}\) can be represented as the diagonal of a parallelogram formed by \(\vec{A}\) and \(\vec{B}\).

### Step 3: Use the Tangent Formula
From the geometry of the situation, we can write the tangent of the angle \(\beta\) (which is \(\frac{\theta}{2}\)) as:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{B \sin(\theta)}{A + B \cos(\theta)} \]
where \(A\) is the magnitude of vector \(\vec{A}\) and \(B\) is the magnitude of vector \(\vec{B}\).

### Step 4: Substitute the Trigonometric Identities
Using the half-angle identities:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \]
Substituting this into our equation gives:
\[ \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{B \sin(\theta)}{A + B \cos(\theta)} \]

### Step 5: Simplify the Equation
Assuming \(\sin(\theta) \neq 0\), we can cancel \(\sin(\theta)\) from both sides:
\[ \frac{1}{1 + \cos(\theta)} = \frac{B}{A + B \cos(\theta)} \]

### Step 6: Cross-Multiply
Cross-multiplying gives:
\[ A + B \cos(\theta) = B(1 + \cos(\theta)) \]

### Step 7: Rearranging the Equation
Rearranging the equation leads to:
\[ A + B \cos(\theta) = B + B \cos(\theta) \]
This simplifies to:
\[ A = B \]

### Conclusion
From our calculations, we find that the magnitudes of the two vectors are equal:
\[ \vec{A} = \vec{B} \]