Sate true/false. The points with position vectors `veca + vecb, veca-vecb and veca +k vecb` are collinear for all real values of k.

Answer

To determine whether the statement is true or false, we need to analyze the position vectors given and check if they are collinear for all real values of \( k \).

### Step-by-step Solution:

1. **Identify the Points**:
Let the points be defined by the following position vectors:
- \( A \) with position vector \( \vec{a} + \vec{b} \)
- \( B \) with position vector \( \vec{a} - \vec{b} \)
- \( C \) with position vector \( \vec{a} + k\vec{b} \)

2. **Find the Vectors \( \vec{AB} \) and \( \vec{AC} \)**:
- The vector \( \vec{AB} \) is given by:
\[ \vec{AB} = \vec{B} - \vec{A} = (\vec{a} - \vec{b}) - (\vec{a} + \vec{b}) = \vec{a} - \vec{b} - \vec{a} - \vec{b} = -2\vec{b} \]
- The vector \( \vec{AC} \) is given by:
\[ \vec{AC} = \vec{C} - \vec{A} = (\vec{a} + k\vec{b}) - (\vec{a} + \vec{b}) = \vec{a} + k\vec{b} - \vec{a} - \vec{b} = (k - 1)\vec{b} \]

3. **Check for Collinearity**:
- For points \( A, B, C \) to be collinear, the vectors \( \vec{AB} \) and \( \vec{AC} \) must be parallel. This means that there exists a scalar \( \lambda \) such that:
\[ \vec{AC} = \lambda \vec{AB} \]
- Substituting the values we found:
\[ (k - 1)\vec{b} = \lambda (-2\vec{b}) \]
- Since \( \vec{b} \) is not the zero vector, we can divide both sides by \( \vec{b} \):
\[ k - 1 = -2\lambda \]

4. **Analyze the Result**:
- The equation \( k - 1 = -2\lambda \) shows that for any real value of \( k \), we can find a corresponding value of \( \lambda \) that satisfies this equation. This indicates that \( A, B, C \) are indeed collinear for all real values of \( k \).

5. **Conclusion**:
- Since we have shown that the points are collinear for all real values of \( k \), the statement is **True**.