If `theta` is the angle between vectors a and b, then `|a xx b| = | a.b|`, then `theta` is equal to

Options

A. 0
B. `180^(@)`
C. `135^(@)`
D. `45^(@)`

Answer

To solve the problem, we need to analyze the given equation involving the magnitudes of the cross product and dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \).

### Step-by-Step Solution:

1. **Understanding the Given Equation:**
We are given that:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a} \cdot \mathbf{b}| \]
Here, \( |\mathbf{a} \times \mathbf{b}| \) represents the magnitude of the cross product of vectors \( \mathbf{a} \) and \( \mathbf{b} \), while \( |\mathbf{a} \cdot \mathbf{b}| \) represents the magnitude of the dot product.

2. **Using the Formulas for Cross Product and Dot Product:**
The magnitudes of the cross product and dot product can be expressed as:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta \]
\[ |\mathbf{a} \cdot \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

3. **Setting the Two Expressions Equal:**
From the given equation, we can substitute the formulas:
\[ |\mathbf{a}| |\mathbf{b}| \sin \theta = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

4. **Dividing Both Sides by \( |\mathbf{a}| |\mathbf{b}| \):**
Assuming \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are non-zero, we can divide both sides by \( |\mathbf{a}| |\mathbf{b}| \):
\[ \sin \theta = \cos \theta \]

5. **Using the Identity of Tangent:**
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Therefore, from \( \sin \theta = \cos \theta \), we can write:
\[ \tan \theta = 1 \]

6. **Finding the Angle \( \theta \):**
The angle \( \theta \) for which \( \tan \theta = 1 \) is:
\[ \theta = 45^\circ \]

### Final Answer:
Thus, the angle \( \theta \) between the vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:
\[ \theta = 45^\circ \]