If a, b, c are non-coplaner vectors such that `bxx c = a, c xx a = b, a xx b = c`, then which of the following is not TRUE?

Options

A. |a|-|b|=0
B. |a| =|b| =|c| =2
C. [a b c ] =1
D. |a||b||c|=1

Answer

To solve the problem, we need to analyze the given vector equations and determine which statement is not true based on the relationships between the vectors \( a \), \( b \), and \( c \).

### Step 1: Analyze the Given Vector Equations
We are given three vector equations:
1. \( b \times c = a \)
2. \( c \times a = b \)
3. \( a \times b = c \)

### Step 2: Understand the Implications of Cross Products
From the properties of cross products, we know that:
- The cross product of two vectors results in a vector that is perpendicular to both of the original vectors.
- The magnitude of the cross product \( |u \times v| = |u||v| \sin(\theta) \), where \( \theta \) is the angle between the two vectors.

### Step 3: Calculate the Magnitudes
Taking the magnitudes of the equations:
1. From \( b \times c = a \):
\[ |a| = |b| |c| \sin(\theta_{bc}) \]
2. From \( c \times a = b \):
\[ |b| = |c| |a| \sin(\theta_{ca}) \]
3. From \( a \times b = c \):
\[ |c| = |a| |b| \sin(\theta_{ab}) \]

### Step 4: Set Up the Relationships
Since \( a \), \( b \), and \( c \) are non-coplanar vectors, we can assume they have the same magnitude. Let's denote:
\[ |a| = |b| = |c| = k \]
Then we can rewrite the equations as:
1. \( k = k^2 \sin(\theta_{bc}) \)
2. \( k = k^2 \sin(\theta_{ca}) \)
3. \( k = k^2 \sin(\theta_{ab}) \)

### Step 5: Analyze the Angles
Since the vectors are non-coplanar, the angles between them are not equal to 0 or 180 degrees. Thus, \( \sin(\theta) \) will not be zero.

### Step 6: Determine the Values of \( k \)
From the equations, we can derive:
\[ 1 = k \sin(\theta_{bc}), \quad 1 = k \sin(\theta_{ca}), \quad 1 = k \sin(\theta_{ab}) \]
This implies:
\[ k = \frac{1}{\sin(\theta_{bc})}, \quad k = \frac{1}{\sin(\theta_{ca})}, \quad k = \frac{1}{\sin(\theta_{ab})} \]

### Step 7: Conclusion on the Magnitudes
Since all three magnitudes are equal, we conclude:
\[ |a| = |b| = |c| = 1 \]

### Step 8: Identify the False Statement
Now, we need to check the options provided in the original question to find which statement is not true based on our calculations.

### Final Answer
After evaluating the relationships and the magnitudes, we find that the statement which is not true among the options provided is option number 2.