If `|vec(a).vec(b)|=|vec(a)xx vec(b)|`, find the angle between `vec(a)` and `vec(b)`.

Answer

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

### Step-by-step Solution:

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

2. **Setting Up the Equation**:
- According to the problem, we have:
\[ |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \]
- Substituting the expressions for the dot product and cross product, we get:
\[ |\ |\vec{a}| |\vec{b}| \cos \theta | = |\ |\vec{a}| |\vec{b}| \sin \theta | \]

3. **Simplifying the Equation**:
- Since \(|\vec{a}|\) and \(|\vec{b}|\) are both positive, we can divide both sides by \(|\vec{a}| |\vec{b}|\):
\[ |\cos \theta| = |\sin \theta| \]

4. **Finding the Angle**:
- The equation \(|\cos \theta| = |\sin \theta|\) implies that \(\cos \theta = \sin \theta\) or \(\cos \theta = -\sin \theta\).
- The angles that satisfy \(\cos \theta = \sin \theta\) are:
\[ \theta = 45^\circ + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]
- The angles that satisfy \(\cos \theta = -\sin \theta\) are:
\[ \theta = 135^\circ + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]

5. **Conclusion**:
- The angles between the vectors \(\vec{a}\) and \(\vec{b}\) that satisfy the original condition are \(45^\circ\) and \(135^\circ\). However, the most common angle considered in vector problems is \(45^\circ\).

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