if `veca xx vecb = vecc ,vecb xx vecc = veca , " where " vecc ne vec0` then (a) `|veca|= |vecc|` (b) `|veca|= |vecb|` (c) `|vecb|=1` (d) `|veca|=|vecb|= |vecc|=1`

Options

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

Explanation

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

Given:
1. \(\vec{a} \times \vec{b} = \vec{c}\)
2. \(\vec{b} \times \vec{c} = \vec{a}\)
3. \(\vec{c} \neq \vec{0}\)

### Step 1: Understand the implications of the cross product
From the first equation, \(\vec{a} \times \vec{b} = \vec{c}\), we know that the magnitude of \(\vec{c}\) can be expressed as:
\[ |\vec{c}| = |\vec{a}| |\vec{b}| \sin \theta \]
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\).

### Step 2: Analyze the angle between \(\vec{a}\) and \(\vec{b}\)
For \(\vec{c}\) to be non-zero, \(\sin \theta\) must be non-zero, which implies that \(\theta\) cannot be \(0\) or \(\pi\). Therefore, \(\theta\) must be \(90^\circ\) (or \(\frac{\pi}{2}\) radians). This means:
\[ |\vec{c}| = |\vec{a}| |\vec{b}| \]

### Step 3: Analyze the second equation
From the second equation, \(\vec{b} \times \vec{c} = \vec{a}\), we can similarly express the magnitude:
\[ |\vec{a}| = |\vec{b}| |\vec{c}| \sin \phi \]
where \(\phi\) is the angle between \(\vec{b}\) and \(\vec{c}\). Since we have established that \(\vec{b}\) and \(\vec{c}\) are also perpendicular (as derived from the conditions), we have \(\phi = 90^\circ\). Thus:
\[ |\vec{a}| = |\vec{b}| |\vec{c}| \]

### Step 4: Set up the equations
Now we have two equations:
1. \( |\vec{c}| = |\vec{a}| |\vec{b}| \)
2. \( |\vec{a}| = |\vec{b}| |\vec{c}| \)

### Step 5: Substitute and simplify
From the first equation, we can substitute \( |\vec{c}| \) into the second equation:
\[ |\vec{a}| = |\vec{b}| (|\vec{a}| |\vec{b}|) \]
This simplifies to:
\[ |\vec{a}| = |\vec{a}| |\vec{b}|^2 \]
If \( |\vec{a}| \neq 0 \) (which it cannot be, since \(\vec{c} \neq \vec{0}\)), we can divide both sides by \( |\vec{a}| \):
\[ 1 = |\vec{b}|^2 \]
Thus, we find that:
\[ |\vec{b}| = 1 \]

### Step 6: Find the relationship between magnitudes
Substituting \( |\vec{b}| = 1 \) back into the first equation:
\[ |\vec{c}| = |\vec{a}| \cdot 1 \implies |\vec{c}| = |\vec{a}| \]

### Conclusion
From our findings, we have:
- \( |\vec{a}| = |\vec{c}| \)
- \( |\vec{b}| = 1 \)

Thus, the correct options are:
(a) \( |\vec{a}| = |\vec{c}| \) (True)
(b) \( |\vec{a}| = |\vec{b}| \) (False)
(c) \( |\vec{b}| = 1 \) (True)
(d) \( |\vec{a}| = |\vec{b}| = |\vec{c}| = 1 \) (False)

### Final Answer
The correct options are (a) and (c).