Translation Vectors 352998

1. **Problem Statement:** We want to understand what translation vectors are and see some examples. 2. **Definition:** A translation vector is a vector that moves every point of a shape or object the same distance in the same direction. 3. **Formula:** If a point has coordinates $P(x, y)$ and the translation vector is $\vec{v} = \langle a, b \rangle$, then the translated point $P'$ is given by: $$P'(x', y') = (x + a, y + b)$$ 4. **Example 1:** Translate the point $P(2, 3)$ by the vector $\vec{v} = \langle 4, -1 \rangle$. Calculation: $$x' = 2 + 4 = 6$$ $$y' = 3 + (-1) = 2$$ So, $P'(6, 2)$. 5. **Example 2:** Translate the point $Q(-1, 5)$ by the vector $\vec{v} = \langle -3, 2 \rangle$. Calculation: $$x' = -1 + (-3) = -4$$ $$y' = 5 + 2 = 7$$ So, $Q'(-4, 7)$. 6. **Explanation:** Translation vectors simply add their components to the original point's coordinates, shifting the point accordingly. 7. **Summary:** Translation vectors move points by adding their components to the point's coordinates, resulting in a shifted position.