The angle between two vector A and B is `theta`. Vector R is the resultant of the two vectors. If R makes an angle `(theta)/(2)` with A, then

Options

A. `A=2B`
B. `A=B//2`
C. `A=B`
D. `AB=1`

Explanation

To solve the problem step by step, we will analyze the relationship between the vectors A, B, and their resultant R given the angle conditions.

### Step 1: Understand the Given Information
We are given:
- Two vectors A and B with an angle θ between them.
- The resultant vector R makes an angle θ/2 with vector A.

### Step 2: Use the Formula for Resultant Vector
The formula for the angle β (which is the angle between the resultant R and vector A) is given by:
\[ \tan \beta = \frac{B \sin \theta}{A + B \cos \theta} \]
In our case, since β = θ/2, we can write:
\[ \tan \left(\frac{\theta}{2}\right) = \frac{B \sin \theta}{A + B \cos \theta} \]

### Step 3: Substitute for sin θ
Using the double angle identity for sine, we can express sin θ as:
\[ \sin \theta = 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \]
Substituting this into our equation gives:
\[ \tan \left(\frac{\theta}{2}\right) = \frac{B \cdot 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)}{A + B \cos \theta} \]

### Step 4: Express tan(θ/2) in terms of sin and cos
We know:
\[ \tan \left(\frac{\theta}{2}\right) = \frac{\sin \left(\frac{\theta}{2}\right)}{\cos \left(\frac{\theta}{2}\right)} \]
So we can rewrite the equation as:
\[ \frac{\sin \left(\frac{\theta}{2}\right)}{\cos \left(\frac{\theta}{2}\right)} = \frac{2B \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)}{A + B \cos \theta} \]

### Step 5: Cancel sin(θ/2) from both sides
Assuming sin(θ/2) ≠ 0, we can cancel it:
\[ \frac{1}{\cos \left(\frac{\theta}{2}\right)} = \frac{2B \cos\left(\frac{\theta}{2}\right)}{A + B \cos \theta} \]

### Step 6: Cross-multiply to simplify
Cross-multiplying gives:
\[ A + B \cos \theta = 2B \cos^2 \left(\frac{\theta}{2}\right) \]

### Step 7: Use the identity for cos θ
Recall that:
\[ \cos \theta = 2 \cos^2 \left(\frac{\theta}{2}\right) - 1 \]
Substituting this into our equation yields:
\[ A + B (2 \cos^2 \left(\frac{\theta}{2}\right) - 1) = 2B \cos^2 \left(\frac{\theta}{2}\right) \]

### Step 8: Rearranging the equation
This simplifies to:
\[ A + B \cdot 2 \cos^2 \left(\frac{\theta}{2}\right) - B = 2B \cos^2 \left(\frac{\theta}{2}\right) \]
Thus:
\[ A - B = 0 \]
This implies:
\[ A = B \]

### Conclusion
The relation between the vectors A and B is:
\[ A = B \]