If `vecaxxvecb=vecc` and `vecbxxvecc=veca` then
a. `veca,vecb,vecc` are orthogonal in pairs and `|veca|=|vecc|,|vecb|=1`
b. `veca,vecb,vecc` are not orthogonal to each other
c. `veca,vecb,vecc` are orthogonal in pairs but `|veca|!=|vecc|`
d. `veca,vecb,vecc` are orthogonal but `|vecb|=1`
OR
If `vecaxxvecb=vecc,vecbxxvecc=veca`, then

Options

A. `|veca|=1,vecb=vecc`
B. `|vecc|=1,|veca|=1`
C. `|vecb|=2,vecc=2veca`
D. `|vecb|=1,|vecc|=|veca|`

Explanation

To solve the problem, we have two equations involving three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):

1. \(\vec{a} \times \vec{b} = \vec{c}\)
2. \(\vec{b} \times \vec{c} = \vec{a}\)

We need to analyze these equations to determine the relationships between the vectors.

### Step 1: Cross Product Properties
Recall the properties of the cross product. For any vectors \(\vec{x}\), \(\vec{y}\), and \(\vec{z}\):
- \(\vec{x} \times \vec{y}\) is orthogonal to both \(\vec{x}\) and \(\vec{y}\).
- The magnitude of the cross product is given by \(|\vec{x} \times \vec{y}| = |\vec{x}||\vec{y}|\sin(\theta)\), where \(\theta\) is the angle between \(\vec{x}\) and \(\vec{y}\).

### Step 2: Analyze \(\vec{a} \times \vec{b} = \vec{c}\)
From the first equation, \(\vec{c}\) is orthogonal to both \(\vec{a}\) and \(\vec{b}\). Therefore:
- \(\vec{a} \cdot \vec{c} = 0\)
- \(\vec{b} \cdot \vec{c} = 0\)

### Step 3: Analyze \(\vec{b} \times \vec{c} = \vec{a}\)
From the second equation, \(\vec{a}\) is orthogonal to both \(\vec{b}\) and \(\vec{c}\). Therefore:
- \(\vec{b} \cdot \vec{a} = 0\)
- \(\vec{c} \cdot \vec{a} = 0\)

### Step 4: Magnitudes of the Vectors
Next, we can find the magnitudes of the vectors. From the first equation:
\[ |\vec{c}| = |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin(\theta) \]
Since \(\vec{a}\) and \(\vec{b}\) are orthogonal, \(\theta = 90^\circ\) and \(\sin(90^\circ) = 1\):
\[ |\vec{c}| = |\vec{a}||\vec{b}| \]

From the second equation:
\[ |\vec{a}| = |\vec{b} \times \vec{c}| = |\vec{b}||\vec{c}|\sin(\phi) \]
Again, since \(\vec{b}\) and \(\vec{c}\) are orthogonal, \(\phi = 90^\circ\) and \(\sin(90^\circ) = 1\):
\[ |\vec{a}| = |\vec{b}||\vec{c}| \]

### Step 5: Setting Up Equations
Now we have two equations:
1. \( |\vec{c}| = |\vec{a}||\vec{b}| \)
2. \( |\vec{a}| = |\vec{b}||\vec{c}| \)

Substituting \( |\vec{c}| \) from the first equation into the second:
\[ |\vec{a}| = |\vec{b}| \cdot (|\vec{a}||\vec{b}|) \]
This simplifies to:
\[ |\vec{a}| = |\vec{a}||\vec{b}|^2 \]
If \( |\vec{a}| \neq 0 \), we can divide both sides by \( |\vec{a}| \):
\[ 1 = |\vec{b}|^2 \]
Thus, we conclude that:
\[ |\vec{b}| = 1 \]

### Step 6: Conclusion on Orthogonality
Since all pairs of vectors are orthogonal, we conclude:
- \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are orthogonal to each other.
- The magnitudes of \(\vec{a}\) and \(\vec{c}\) are equal.

### Final Answer
Thus, the correct option is:
**a. \(\vec{a}, \vec{b}, \vec{c}\) are orthogonal in pairs and \(|\vec{a}| = |\vec{c}|, |\vec{b}| = 1\)**.