If `theta` is the angle between any two vectors `veca and vecb`, then `|veca.vecb|=|veca xx vecb|` when `theta` is equal to

Options

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

Answer

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 equals the magnitude of their cross product.

### Step-by-step Solution:

1. **Write the expressions for dot product and cross product**:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]

2. **Set the magnitudes of the dot product and cross product equal**:
\[ |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \]
This leads to:
\[ |\vec{a}| |\vec{b}| \cos \theta = |\vec{a}| |\vec{b}| \sin \theta \]

3. **Assuming \( |\vec{a}| \) and \( |\vec{b}| \) are not zero, we can divide both sides by \( |\vec{a}| |\vec{b}| \)**:
\[ \cos \theta = \sin \theta \]

4. **Rearranging the equation**:
\[ \frac{\cos \theta}{\sin \theta} = 1 \]
This implies:
\[ \cot \theta = 1 \]

5. **Finding the angle \( \theta \)**:
The angle \( \theta \) for which \( \cot \theta = 1 \) is:
\[ \theta = \cot^{-1}(1) = \frac{\pi}{4} \]

6. **Conclusion**:
Therefore, the angle \( \theta \) between the vectors \( \vec{a} \) and \( \vec{b} \) is:
\[ \theta = \frac{\pi}{4} \]