If the magnitude of `vec(a) xx vec(b)` equals to `vec(a).vec(b)` then which one of the following is correct?

Options

A. `vec(a)= vec(b)`
B. The angle between `vec(a ) and vec(b)` a is `45^(@)`
C. `vec(a)` is parallel to `vec(b)`
D. `vec(a)` is perpendicular to `vec(b)`

Answer

To solve the problem, we need to analyze the relationship between the cross product and the dot product of two vectors \(\vec{a}\) and \(\vec{b}\).

### Step-by-step Solution:

1. **Understanding the Given Condition**:
We are given that the magnitude of the cross product of \(\vec{a}\) and \(\vec{b}\) is equal to the dot product of \(\vec{a}\) and \(\vec{b}\):
\[ |\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b} \]

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

3. **Expressing the Dot Product**:
The dot product is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

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

5. **Dividing by \(|\vec{a}| |\vec{b}|\)**:
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 Trigonometric Identity**:
The equation \(\sin \theta = \cos \theta\) can be rewritten using the tangent function:
\[ \frac{\sin \theta}{\cos \theta} = 1 \implies \tan \theta = 1 \]

7. **Finding the Angle**:
The angle \(\theta\) for which \(\tan \theta = 1\) is:
\[ \theta = 45^\circ \]

8. **Conclusion**:
Therefore, the correct option is that the angle between the two vectors \(\vec{a}\) and \(\vec{b}\) is \(45^\circ\).

### Final Answer:
The correct option is **Option B: The angle between them is 45 degrees.**