If `vec(a)` and `vec(b)` are two vectors such that `|vec(a) xx vec(b)| = vec(a).vec(b)`, then what is the angle between `vec(a)` and `vec(b)`.

Answer

To solve the problem, we need to find the angle between the two vectors \(\vec{a}\) and \(\vec{b}\) given that \(|\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b}\).

### Step-by-step Solution:

1. **Understanding the given condition**:
We know that:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
and
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
where \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\).

2. **Setting up the equation**:
Given that \(|\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b}\), we can substitute the expressions for the cross product and dot product:
\[ |\vec{a}| |\vec{b}| \sin \theta = |\vec{a}| |\vec{b}| \cos \theta \]

3. **Dividing both sides**:
Assuming \(|\vec{a}| \neq 0\) and \(|\vec{b}| \neq 0\), we can divide both sides by \(|\vec{a}| |\vec{b}|\):
\[ \sin \theta = \cos \theta \]

4. **Using the identity**:
We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, we can write:
\[ \tan \theta = 1 \]

5. **Finding the angle**:
The angle \(\theta\) for which \(\tan \theta = 1\) is:
\[ \theta = 45^\circ \quad \text{or} \quad \theta = \frac{\pi}{4} \text{ radians} \]

### Final Answer:
The angle between the vectors \(\vec{a}\) and \(\vec{b}\) is \(45^\circ\).