The points with position vectors `vecx + vecy, vecx-vecy and vecx +λ vecy` are collinear for all real values of λ.

Explanation

To determine whether the points with position vectors \(\vec{x} + \vec{y}\), \(\vec{x} - \vec{y}\), and \(\vec{x} + \lambda \vec{y}\) are collinear for all real values of \(\lambda\), we can follow these steps:

### Step 1: Define the Points
Let:
- Point A be represented by the position vector \(\vec{A} = \vec{x} + \vec{y}\)
- Point B be represented by the position vector \(\vec{B} = \vec{x} - \vec{y}\)
- Point C be represented by the position vector \(\vec{C} = \vec{x} + \lambda \vec{y}\)

### Step 2: Find Vectors AB and BC
To check for collinearity, we need to find the vectors \(\vec{AB}\) and \(\vec{BC}\).

1. **Vector AB**:
\[ \vec{AB} = \vec{B} - \vec{A} = (\vec{x} - \vec{y}) - (\vec{x} + \vec{y}) \]
Simplifying this:
\[ \vec{AB} = \vec{x} - \vec{y} - \vec{x} - \vec{y} = -2\vec{y} \]

2. **Vector BC**:
\[ \vec{BC} = \vec{C} - \vec{B} = (\vec{x} + \lambda \vec{y}) - (\vec{x} - \vec{y}) \]
Simplifying this:
\[ \vec{BC} = \vec{x} + \lambda \vec{y} - \vec{x} + \vec{y} = (\lambda + 1)\vec{y} \]

### Step 3: Check for Parallelism
For points A, B, and C to be collinear, the vectors \(\vec{AB}\) and \(\vec{BC}\) must be parallel.

From our calculations:
- \(\vec{AB} = -2\vec{y}\)
- \(\vec{BC} = (\lambda + 1)\vec{y}\)

Both vectors are scalar multiples of \(\vec{y}\). This means that:
\[ \vec{AB} \text{ is parallel to } \vec{BC} \text{ for all real values of } \lambda \]

### Conclusion
Since \(\vec{AB}\) and \(\vec{BC}\) are parallel for all values of \(\lambda\), the points A, B, and C are collinear for all real values of \(\lambda\).

Thus, the statement is **true**.

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