`vec(A)` and `vec(B)` are two Vectors and `theta` is the angle between them, if `|vec(A)xxvec(B)|= sqrt(3)(vec(A).vec(B))` the value of `theta` is

Options

A. `60^(@)`
B. `45^(@)`
C. `30^(@)`
D. `90^(@)`

Answer

To solve the problem, we need to use the relationship between the cross product and dot product of two vectors.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We are given that:
\[ |\vec{A} \times \vec{B}| = \sqrt{3} (\vec{A} \cdot \vec{B}) \]
where \(\theta\) is the angle between the vectors \(\vec{A}\) and \(\vec{B}\).

2. **Using the Formulas for Cross Product and Dot Product:**
The magnitude of the cross product of two vectors is given by:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]
The dot product of two vectors is given by:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]

3. **Substituting the Formulas into the Given Equation:**
Substitute the formulas into the given equation:
\[ |\vec{A}| |\vec{B}| \sin \theta = \sqrt{3} (|\vec{A}| |\vec{B}| \cos \theta) \]

4. **Canceling Common Terms:**
Assuming \(|\vec{A}| \neq 0\) and \(|\vec{B}| \neq 0\), we can cancel \(|\vec{A}| |\vec{B}|\) from both sides:
\[ \sin \theta = \sqrt{3} \cos \theta \]

5. **Dividing Both Sides by \(\cos \theta\):**
We can divide both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)):
\[ \frac{\sin \theta}{\cos \theta} = \sqrt{3} \]
This simplifies to:
\[ \tan \theta = \sqrt{3} \]

6. **Finding the Angle \(\theta\):**
The angle \(\theta\) for which \(\tan \theta = \sqrt{3}\) is:
\[ \theta = 60^\circ \]

### Conclusion:
Thus, the value of \(\theta\) is \(60^\circ\).