If `(vecaxxvecb)xx(vecbxxvecc)=vecb, where veca,vecb and vecc` are non zero vectors then (A) `veca,vecb and vecc can be coplanar (B) `veca,vecb and vecc` must be coplanar (C) `veca,vecb and vecc cannot be coplanar (D) none of these

Options

A. `veca,vecb and vecv` can be coplanar
B. `veca, vecb and vecc` must be coplanar
C. `veca,vecb and vecc` cannot be coplanar
D. none of these

Explanation

To solve the problem, we need to analyze the equation given:

\[ (\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) = \vec{b} \]

where \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are non-zero vectors. We will use the properties of vector products to determine the relationship between these vectors, particularly focusing on their coplanarity.

### Step 1: Apply the Vector Triple Product Identity
We start by applying the vector triple product identity, which states that:

\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]

In our case, let \(\vec{x} = \vec{a}\), \(\vec{y} = \vec{b}\), and \(\vec{z} = \vec{c}\). Thus, we can rewrite:

\[ (\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 2: Set the Equation
Now, we set this equal to \(\vec{b}\):

\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} = \vec{b} \]

### Step 3: Rearranging the Equation
Rearranging gives us:

\[ (\vec{a} \cdot \vec{c}) \vec{b} - \vec{b} = (\vec{a} \cdot \vec{b}) \vec{c} \]

This simplifies to:

\[ (\vec{a} \cdot \vec{c} - 1) \vec{b} = (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 4: Analyzing the Coefficients
Since \(\vec{b}\) is a non-zero vector, we can analyze the coefficients:

1. If \(\vec{a} \cdot \vec{c} - 1 = 0\), then \(\vec{a} \cdot \vec{c} = 1\).
2. If \(\vec{a} \cdot \vec{b} \neq 0\), then \(\vec{c}\) must be a non-zero vector.

### Step 5: Coplanarity Condition
For vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) to be coplanar, the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) must be zero. However, we have shown that:

\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 1 \]

This indicates that the vectors are not coplanar.

### Conclusion
Based on the analysis, we conclude that:

**The correct answer is (C) \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) cannot be coplanar.**

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