what is the angle between `veca and vecb, if veca. vecb = 3 and | veca xx vecb| = 3 sqrt3.`

Answer

To find the angle between the vectors \(\vec{A}\) and \(\vec{B}\) given the conditions \(\vec{A} \cdot \vec{B} = 3\) and \(|\vec{A} \times \vec{B}| = 3\sqrt{3}\), we can follow these steps:

### Step 1: Write the equations for the dot product and cross product
The dot product of two vectors is given by:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]
The magnitude of the cross product is given by:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]

### Step 2: Set up the equations
From the given information, we can set up the following equations:
1. \( |\vec{A}| |\vec{B}| \cos \theta = 3 \) (Equation 1)
2. \( |\vec{A}| |\vec{B}| \sin \theta = 3\sqrt{3} \) (Equation 2)

### Step 3: Divide Equation 2 by Equation 1
To eliminate the magnitudes of the vectors, we can divide Equation 2 by Equation 1:
\[ \frac{|\vec{A}| |\vec{B}| \sin \theta}{|\vec{A}| |\vec{B}| \cos \theta} = \frac{3\sqrt{3}}{3} \]
This simplifies to:
\[ \tan \theta = \sqrt{3} \]

### Step 4: Find the angle \(\theta\)
The angle \(\theta\) for which \(\tan \theta = \sqrt{3}\) is:
\[ \theta = \frac{\pi}{3} \quad \text{or} \quad 60^\circ \]

### Conclusion
Thus, the angle between the vectors \(\vec{A}\) and \(\vec{B}\) is:
\[ \theta = \frac{\pi}{3} \text{ radians or } 60^\circ \]

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