If `vec(A),vec(B),vec(C)` are mutually perpendicular vectors then which of the following statements is wrong?

Options

A. `vec(C) xx (vec(A) xxvec(B))=0`
B. `(vec(A)xx vec(B))/(|vec(A)xxvec(B)|)=(vec(C))/(|vec(C)|)`
C. `vec(A).vec(B) = vec(B).vec(C) =vec(C).vec(A)=0`
D. `(vec(B)+vec(C))` is perpendicular to ` vec(A)`

Answer

To determine which statement is wrong regarding the mutually perpendicular vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\), we need to analyze the properties of these vectors and their relationships.

### Step-by-Step Solution:

1. **Understanding Mutually Perpendicular Vectors**:
- Vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) are said to be mutually perpendicular if the angle between any two of them is \(90^\circ\). This means:
\[ \vec{A} \cdot \vec{B} = 0, \quad \vec{B} \cdot \vec{C} = 0, \quad \vec{C} \cdot \vec{A} = 0 \]

2. **Cross Product of Two Vectors**:
- The cross product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by:
\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \hat{n} \]
where \(\theta\) is the angle between them and \(\hat{n}\) is the unit vector perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\). Since \(\vec{A}\) and \(\vec{B}\) are perpendicular, \(\theta = 90^\circ\) and \(\sin(90^\circ) = 1\), thus:
\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \hat{C} \]
where \(\hat{C}\) is in the direction of \(\vec{C}\).

3. **Analyzing the Statements**:
- **Statement 1**: \(\vec{A} \times \vec{B}\) is perpendicular to both \(\vec{A}\) and \(\vec{B}\). This is true.
- **Statement 2**: The cross product \(\vec{A} \times \vec{B}\) is equal to \(\vec{C}\) or \(-\vec{C}\). This is also true, but we must consider the direction.
- **Statement 3**: The dot product of any two mutually perpendicular vectors is zero. This is true.
- **Statement 4**: The vector \(\vec{B} + \vec{C}\) is perpendicular to \(\vec{A}\). This is true because \(\vec{B}\) and \(\vec{C}\) lie in a plane perpendicular to \(\vec{A}\).

4. **Identifying the Wrong Statement**:
- The statement that \(\frac{\vec{A} \times \vec{B}}{|\vec{A} \times \vec{B}|} = \frac{\vec{C}}{|\vec{C}|}\) is incorrect because it assumes that the direction of \(\vec{A} \times \vec{B}\) is the same as that of \(\vec{C}\). However, \(\vec{A} \times \vec{B}\) could also point in the opposite direction (i.e., \(-\vec{C}\)). Therefore, this statement is wrong.

### Conclusion:
The wrong statement is:
\[ \text{Option B: } \frac{\vec{A} \times \vec{B}}{|\vec{A} \times \vec{B}|} = \frac{\vec{C}}{|\vec{C}|} \]