If a, b and c are any three vectors, then `a times (b times c)=(a times b) times c` if and only if

Options

A. b and c are collinear
B. a and c are collinear
C. a and b are collinera
D. none of these

Answer

To solve the problem, we need to determine the conditions under which the vector identity \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \) holds true.

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The expression \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) can be simplified using the vector triple product identity:
\[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \]
Applying this identity, we have:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]

2. **Applying the Triple Product Identity Again**:
Now, we apply the triple product identity to the right-hand side \( (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \):
\[ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b} \]

3. **Setting the Two Expressions Equal**:
We now set the two expressions equal to each other:
\[ (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b} \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} + (\mathbf{c} \cdot \mathbf{a}) \mathbf{b} = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} + (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]
This can be simplified to:
\[ (\mathbf{a} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) \mathbf{b} = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} + (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]

5. **Analyzing Collinearity**:
For the equality to hold, the vectors must be collinear. This means that:
- \( \mathbf{b} \) must be a scalar multiple of \( \mathbf{c} \)
- \( \mathbf{a} \) must be a scalar multiple of \( \mathbf{b} \)
- \( \mathbf{a} \) must be a scalar multiple of \( \mathbf{c} \)

6. **Conclusion**:
Thus, the condition for \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \) to hold true is that at least two of the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) must be collinear.

### Final Answer:
The expression \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \) holds if and only if at least two of the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are collinear.