`"|"oversetrightarrow(a).oversetrightarrow(b)"|"=sqrt3"|"oversetrightarrow(a)xxoversetrightarrow(b)"|"`, then the angle between `oversetrightarrow(a)andoversetrightarrow(b)` is:

Options

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

Answer

To solve the problem, we need to find the angle between the vectors \( \oversetrightarrow{a} \) and \( \oversetrightarrow{b} \) given the relationship:

\[ |\oversetrightarrow{a} \cdot \oversetrightarrow{b}| = \sqrt{3} |\oversetrightarrow{a} \times \oversetrightarrow{b}| \]

### Step-by-step Solution:

1. **Understanding the Dot and Cross Products**:
- The dot product of two vectors \( \oversetrightarrow{a} \) and \( \oversetrightarrow{b} \) is given by:
\[ \oversetrightarrow{a} \cdot \oversetrightarrow{b} = |\oversetrightarrow{a}| |\oversetrightarrow{b}| \cos \theta \]
- The cross product of two vectors \( \oversetrightarrow{a} \) and \( \oversetrightarrow{b} \) is given by:
\[ \oversetrightarrow{a} \times \oversetrightarrow{b} = |\oversetrightarrow{a}| |\oversetrightarrow{b}| \sin \theta \]

2. **Substituting the Products into the Given Equation**:
- We can substitute the expressions for the dot and cross products into the given equation:
\[ |\oversetrightarrow{a}| |\oversetrightarrow{b}| \cos \theta = \sqrt{3} |\oversetrightarrow{a}| |\oversetrightarrow{b}| \sin \theta \]

3. **Cancelling Common Terms**:
- Assuming \( |\oversetrightarrow{a}| \) and \( |\oversetrightarrow{b}| \) are not zero, we can cancel these terms from both sides:
\[ \cos \theta = \sqrt{3} \sin \theta \]

4. **Rearranging the Equation**:
- We can rearrange this equation to express it in terms of tangent:
\[ \frac{\cos \theta}{\sin \theta} = \sqrt{3} \]
- This simplifies to:
\[ \cot \theta = \sqrt{3} \]

5. **Finding the Angle**:
- Taking the reciprocal gives us:
\[ \tan \theta = \frac{1}{\sqrt{3}} \]
- We know that \( \tan \theta = \frac{1}{\sqrt{3}} \) corresponds to an angle of:
\[ \theta = \frac{\pi}{6} \text{ or } 30^\circ \]

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

### Final Answer:
The angle between \( \oversetrightarrow{a} \) and \( \oversetrightarrow{b} \) is \( \frac{\pi}{6} \) radians or \( 30^\circ \).