Three points with position vectors, a, b, c are collinear if

Options

A. `a xx b + b xx c + c xx a = 0`
B. `a, b + b .c + c . A =0`
C. `a. (b xx c) = 0`
D. `a + b + c = 0`

Answer

To determine the condition for three points with position vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) to be collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Collinearity**:
Three points \( A, B, C \) are collinear if the vectors \( \mathbf{AB} \) and \( \mathbf{AC} \) are parallel. This can be expressed mathematically as:
\[ \mathbf{AB} = \mathbf{b} - \mathbf{a} \]
\[ \mathbf{AC} = \mathbf{c} - \mathbf{a} \]

2. **Using Cross Product**:
The vectors \( \mathbf{AB} \) and \( \mathbf{AC} \) are parallel if their cross product is zero:
\[ \mathbf{AB} \times \mathbf{AC} = \mathbf{0} \]

3. **Substituting the Vectors**:
Substitute the expressions for \( \mathbf{AB} \) and \( \mathbf{AC} \):
\[ (\mathbf{b} - \mathbf{a}) \times (\mathbf{c} - \mathbf{a}) = \mathbf{0} \]

4. **Expanding the Cross Product**:
Using the distributive property of the cross product:
\[ \mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{a} - \mathbf{a} \times \mathbf{c} + \mathbf{a} \times \mathbf{a} = \mathbf{0} \]
Since the cross product of any vector with itself is zero (\( \mathbf{a} \times \mathbf{a} = \mathbf{0} \)), we simplify to:
\[ \mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{a} - \mathbf{a} \times \mathbf{c} = \mathbf{0} \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ \mathbf{b} \times \mathbf{c} = \mathbf{b} \times \mathbf{a} + \mathbf{a} \times \mathbf{c} \]

6. **Final Condition**:
The final condition for the points \( A, B, C \) to be collinear can be expressed as:
\[ \mathbf{b} \times \mathbf{c} = \mathbf{0} \]
or equivalently:
\[ \mathbf{c} - \mathbf{a} = k(\mathbf{b} - \mathbf{a}) \quad \text{for some scalar } k \]

### Conclusion:
Thus, the three points with position vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are collinear if:
\[ \mathbf{b} \times \mathbf{c} = \mathbf{0} \]