If `theta` is the angle between any two vectors ` veca` and ` vecb` , then `| vecadot vecb|=| vecaxx vecb|` when `theta` is equal to (a) 0 (B) `pi/4` (C) `pi/2` (d) `pi`

Options

A. 0
B. `(pi)/(4)`
C. `(pi)/(2)`
D. `pi`

Explanation

To solve the problem, we need to find the angle \( \theta \) between two vectors \( \vec{A} \) and \( \vec{B} \) such that the magnitude of their dot product is equal to the magnitude of their cross product.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ |\vec{A} \cdot \vec{B}| = |\vec{A}| |\vec{B}| \cos \theta \]
- The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]

2. **Set Up the Equation**:
- According to the problem, we have:
\[ |\vec{A} \cdot \vec{B}| = |\vec{A} \times \vec{B}| \]
- Substituting the formulas for dot and cross products, we get:
\[ |\vec{A}| |\vec{B}| \cos \theta = |\vec{A}| |\vec{B}| \sin \theta \]

3. **Simplify the Equation**:
- Assuming \( |\vec{A}| \) and \( |\vec{B}| \) are not zero, we can divide both sides by \( |\vec{A}| |\vec{B}| \):
\[ \cos \theta = \sin \theta \]

4. **Use Trigonometric Identity**:
- We can rewrite the equation as:
\[ \frac{\sin \theta}{\cos \theta} = 1 \]
- This simplifies to:
\[ \tan \theta = 1 \]

5. **Find the Angle**:
- The angle \( \theta \) for which \( \tan \theta = 1 \) is:
\[ \theta = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]
- Since we are looking for angles in the range of \( [0, 2\pi] \), the principal solution is:
\[ \theta = \frac{\pi}{4} \]

6. **Conclusion**:
- The value of \( \theta \) that satisfies the given condition is:
\[ \boxed{\frac{\pi}{4}} \]