1. **Stating the problem:** We have an original line given by the equation $y = x + 1$ and its transformed image line $y = x + 2$. We need to find the transforming vector that moves the original line to the transformed line. 2. **Understanding the transformation:** The transformation shifts the line vertically from $y = x + 1$ to $y = x + 2$. This means every point on the original line is moved up by some amount. 3. **Formula and explanation:** A transforming vector $\vec{v} = (a, b)$ moves every point $(x, y)$ on the original line to $(x + a, y + b)$ on the transformed line. 4. **Finding the vector components:** Since the slope of the line remains the same (both lines have slope 1), the horizontal component $a$ of the vector is 0 (no horizontal shift). 5. **Vertical shift:** The $y$-intercept changes from 1 to 2, so the vertical component $b$ is $2 - 1 = 1$. 6. **Conclusion:** The transforming vector is $$\vec{v} = (0, 1)$$ This means the line is shifted vertically upward by 1 unit.