If `|vec(A) xx vec(B)| = |vec(A).vec(B)|`, then the angle between `vec(A)` and `vec(B)` will be :

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

To solve the problem, we start with the given equation:

\[ |\vec{A} \times \vec{B}| = |\vec{A} \cdot \vec{B}| \]

### Step 1: Write the expressions for the magnitudes of the cross product and the dot product.

The magnitude of the cross product \(|\vec{A} \times \vec{B}|\) 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 \(\vec{B}\).

The magnitude of the dot product \(|\vec{A} \cdot \vec{B}|\) is given by:

\[ |\vec{A} \cdot \vec{B}| = |\vec{A}| |\vec{B}| \cos \theta \]

### Step 2: Set the two expressions equal to each other.

From the problem statement, we have:

\[ |\vec{A}| |\vec{B}| \sin \theta = |\vec{A}| |\vec{B}| \cos \theta \]

### Step 3: Simplify the equation.

Assuming \(|\vec{A}| \neq 0\) and \(|\vec{B}| \neq 0\), we can divide both sides by \(|\vec{A}| |\vec{B}|\):

\[ \sin \theta = \cos \theta \]

### Step 4: Solve for \(\theta\).

The equation \(\sin \theta = \cos \theta\) can be rewritten as:

\[ \tan \theta = 1 \]

The angle \(\theta\) that satisfies this equation is:

\[ \theta = 45^\circ \]

### Conclusion

Thus, the angle between \(\vec{A}\) and \(\vec{B}\) is:

\[ \theta = 45^\circ \]