If `veca and vecb` are two vectors such that `|veca xx vecb|= veca.vecb ,` then what is the angle between `veca and vecb` ?

Answer

To solve the problem, we need to find the angle between two vectors \( \vec{a} \) and \( \vec{b} \) given that the magnitude of their cross product is equal to the dot product of the vectors.

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We are given that:
\[ |\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b} \]

2. **Using the Formula for Cross Product**:
The magnitude of the cross product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
where \( \theta \) is the angle between the vectors \( \vec{a} \) and \( \vec{b} \).

3. **Using the Formula for Dot Product**:
The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

4. **Setting the Two Expressions Equal**:
From the condition given in the problem, we can set the two expressions equal to each other:
\[ |\vec{a}| |\vec{b}| \sin \theta = |\vec{a}| |\vec{b}| \cos \theta \]

5. **Canceling Common Terms**:
Assuming \( |\vec{a}| \) and \( |\vec{b}| \) are not zero, we can divide both sides by \( |\vec{a}| |\vec{b}| \):
\[ \sin \theta = \cos \theta \]

6. **Using the Identity of Tangent**:
We can rewrite the equation as:
\[ \frac{\sin \theta}{\cos \theta} = 1 \]
This implies:
\[ \tan \theta = 1 \]

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

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

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