If `vec(a)xx vec(b) = vec(c) and vec(b) xx vec(c) = vec(a)`, then which one of the following is correct?

Options

A. `vec(a),vec(b),vec(c)` are orthogonal in pairs and `|vec(a)|=|vec(c)| and |vec(b)|=1`
B. `vec(a),vec(b),vec(c)` are non-orthogonal to each other
C. `vec(a), vec(b), vec(c)` are orthogonal in pairs but `|vec(a)|!=|vec(c)|`
D. `vec(a), vec(b), vec(c)` are orthogonal is pairs but `|vec(b)|!=1`

Explanation

To solve the problem, we need to analyze the given vector equations step by step.

### Step-by-Step Solution:

1. **Given Equations**:
We are given two vector equations:
\[ \vec{A} \times \vec{B} = \vec{C} \]
\[ \vec{B} \times \vec{C} = \vec{A} \]

2. **Understanding Cross Product**:
The cross product of two vectors results in a vector that is perpendicular to both of the original vectors. Therefore, from the first equation, we can conclude:
- \(\vec{C}\) is perpendicular to both \(\vec{A}\) and \(\vec{B}\).

3. **Analyzing the Second Equation**:
From the second equation, we have:
- \(\vec{A}\) is perpendicular to both \(\vec{B}\) and \(\vec{C}\).

This means that \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) are mutually perpendicular to each other.

4. **Conclusion on Orthogonality**:
Since all three vectors are mutually perpendicular, we can conclude that:
- \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) are orthogonal in pairs.

5. **Finding Magnitudes**:
Now, let's find the magnitudes of these vectors. We know:
\[ |\vec{A} \times \vec{B}| = |\vec{C}| \]
The magnitude of the cross product can be expressed as:
\[ |\vec{A}||\vec{B}|\sin(\theta) \]
where \(\theta\) is the angle between \(\vec{A}\) and \(\vec{B}\). Since \(\vec{A}\) and \(\vec{B}\) are perpendicular, \(\theta = 90^\circ\) and \(\sin(90^\circ) = 1\). Thus:
\[ |\vec{C}| = |\vec{A}||\vec{B}| \]

6. **Applying the Second Equation**:
Similarly, from the second equation:
\[ |\vec{B} \times \vec{C}| = |\vec{A}| \]
Again, since \(\vec{B}\) and \(\vec{C}\) are perpendicular, we have:
\[ |\vec{A}| = |\vec{B}||\vec{C}| \]

7. **Equating the Two Magnitude Relations**:
Now we have two equations:
\[ |\vec{C}| = |\vec{A}||\vec{B}| \]
\[ |\vec{A}| = |\vec{B}||\vec{C}| \]
Substituting the first equation into the second gives:
\[ |\vec{A}| = |\vec{B}||\vec{A}||\vec{B}| \]
This simplifies to:
\[ 1 = |\vec{B}|^2 \]
Therefore, we find:
\[ |\vec{B}| = 1 \]

8. **Finding the Relationship Between A and C**:
Substituting \(|\vec{B}| = 1\) back into the first equation:
\[ |\vec{C}| = |\vec{A}| \cdot 1 = |\vec{A}| \]
Thus, we conclude:
\[ |\vec{A}| = |\vec{C}| \]

### Final Conclusion:
From the analysis, we conclude that:
- \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) are orthogonal in pairs.
- \(|\vec{A}| = |\vec{C}|\)
- \(|\vec{B}| = 1\)

Thus, the correct option is:
**A: \(\vec{A}, \vec{B}, \vec{C}\) are orthogonal in pairs and \(|\vec{A}| = |\vec{C}|\) and \(|\vec{B}| = 1\)**.