Suppose a and b are two non-zero vectors and angle between a and b is `theta, " where "0 lt theta lt pi//2`. If `abs(a times b)=abs(a*b)," then "pi//theta=`_______

Answer

To solve the problem, we need to find the value of \( \frac{\pi}{\theta} \) given that the magnitudes of the cross product and the dot product of two non-zero vectors \( \mathbf{a} \) and \( \mathbf{b} \) are equal, and the angle \( \theta \) between them is in the range \( 0 < \theta < \frac{\pi}{2} \).

### Step-by-step Solution:

1. **Write the expressions for the cross product and dot product:**
- The magnitude of the cross product is given by:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta \]
- The dot product is given by:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

2. **Set the magnitudes equal:**
Since the magnitudes of the cross product and the dot product are equal, we have:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a} \cdot \mathbf{b}| \]
This gives us the equation:
\[ |\mathbf{a}| |\mathbf{b}| \sin \theta = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

3. **Cancel the common terms:**
Since \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are both non-zero, we can divide both sides by \( |\mathbf{a}| |\mathbf{b}| \):
\[ \sin \theta = \cos \theta \]

4. **Solve for \( \theta \):**
The equation \( \sin \theta = \cos \theta \) holds true when:
\[ \theta = \frac{\pi}{4} \]

5. **Calculate \( \frac{\pi}{\theta} \):**
Now, substituting \( \theta = \frac{\pi}{4} \) into \( \frac{\pi}{\theta} \):
\[ \frac{\pi}{\theta} = \frac{\pi}{\frac{\pi}{4}} = 4 \]

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