Two vectors `A` and `B` are such that `A+B = C` and `A^2 +B^2 = C^2`. If `theta` is the angle between positive direction of `A` and `B`, then the correct statement is

Options

A. `theta = pi`
B. `theta = (2pi)/(3)`
C. `theta = 0`
D. `theta = (pi)/(2)`

Explanation

To solve the problem, we start with the given equations:

1. **Equation 1**: \( A + B = C \)
2. **Equation 2**: \( A^2 + B^2 = C^2 \)

We need to find the angle \( \theta \) between the vectors \( A \) and \( B \).

### Step 1: Expand the expression for \( C^2 \)

Using the first equation, we can express \( C \) in terms of \( A \) and \( B \):
\[ C = A + B \]
Now, squaring both sides gives:
\[ C^2 = (A + B)^2 \]
Expanding this using the formula for the square of a sum:
\[ C^2 = A^2 + B^2 + 2AB \cos \theta \]

### Step 2: Substitute \( C^2 \) into the second equation

From the second equation, we know:
\[ A^2 + B^2 = C^2 \]
Substituting the expanded expression for \( C^2 \) into this equation gives:
\[ A^2 + B^2 = A^2 + B^2 + 2AB \cos \theta \]

### Step 3: Simplify the equation

By subtracting \( A^2 + B^2 \) from both sides, we get:
\[ 0 = 2AB \cos \theta \]

### Step 4: Analyze the equation

For the equation \( 0 = 2AB \cos \theta \) to hold true, either \( A \) or \( B \) must be zero, or \( \cos \theta \) must be zero. Since we are interested in the angle \( \theta \), we focus on:
\[ \cos \theta = 0 \]

### Step 5: Determine the angle \( \theta \)

The cosine of an angle is zero at specific angles:
- \( \theta = \frac{\pi}{2} \) (90 degrees)
- \( \theta = \frac{3\pi}{2} \) (270 degrees)

However, since we are looking for the angle between the positive directions of \( A \) and \( B \), we consider:
\[ \theta = \frac{\pi}{2} \]

### Conclusion

Thus, the correct statement is that the angle \( \theta \) between the positive directions of \( A \) and \( B \) is:
\[ \theta = \frac{\pi}{2} \]