1. **Problem Statement:** We are given a vector $v = (1,1)$ in $\mathbb{R}^2$. We want to show that reflecting $v$ about the $y$-axis produces the same result as rotating $v$ by 90° counterclockwise about the origin. 2. **Reflection about the $y$-axis:** The reflection of a vector $(x,y)$ about the $y$-axis changes the $x$-coordinate to its negative while keeping the $y$-coordinate the same. The formula is: $$\text{Reflection}(x,y) = (-x, y)$$ Applying this to $v = (1,1)$: $$(-1, 1)$$ 3. **Rotation by 90° counterclockwise:** The rotation of a vector $(x,y)$ by 90° counterclockwise about the origin is given by: $$\text{Rotation}_{90^\circ}(x,y) = (-y, x)$$ Applying this to $v = (1,1)$: $$(-1, 1)$$ 4. **Comparison:** Both the reflection about the $y$-axis and the 90° counterclockwise rotation of $v$ yield the vector $(-1,1)$. 5. **Conclusion:** Therefore, for the vector $v = (1,1)$, reflecting about the $y$-axis produces the same result as rotating by 90° counterclockwise about the origin.