If `theta` is the angle between any two vectors `bara` and `barb` and `| bara xx barb | = | bara.barb |`, then value of `theta` is

Options

A. `(pi)/(3)`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

To solve the problem, we start with the given condition that the magnitude of the cross product of two vectors \( \bar{a} \) and \( \bar{b} \) is equal to the magnitude of their dot product.

### Step-by-Step Solution:

1. **Write the given condition**:
\[ |\bar{a} \times \bar{b}| = |\bar{a} \cdot \bar{b}| \]

2. **Express the magnitude of the cross product**:
The magnitude of the cross product of two vectors can be expressed as:
\[ |\bar{a} \times \bar{b}| = |\bar{a}| |\bar{b}| \sin \theta \]

3. **Express the magnitude of the dot product**:
The magnitude of the dot product of two vectors can be expressed as:
\[ |\bar{a} \cdot \bar{b}| = |\bar{a}| |\bar{b}| \cos \theta \]

4. **Set the two expressions equal**:
From the given condition, we can set the two expressions equal to each other:
\[ |\bar{a}| |\bar{b}| \sin \theta = |\bar{a}| |\bar{b}| \cos \theta \]

5. **Cancel out the common terms**:
Assuming \( |\bar{a}| \) and \( |\bar{b}| \) are not zero, we can divide both sides by \( |\bar{a}| |\bar{b}| \):
\[ \sin \theta = \cos \theta \]

6. **Use the identity for tangent**:
We can rewrite the equation as:
\[ \frac{\sin \theta}{\cos \theta} = 1 \]
This simplifies to:
\[ \tan \theta = 1 \]

7. **Find the angle \( \theta \)**:
The angle \( \theta \) for which \( \tan \theta = 1 \) is:
\[ \theta = \frac{\pi}{4} \text{ (or 45 degrees)} \]

### Final Answer:
Thus, the value of \( \theta \) is:
\[ \theta = \frac{\pi}{4} \]