1. **Stating the problem:** We will explore vector translation, vector addition and subtraction both graphically and algebraically, scalar multiplication of vectors, calculation of vector magnitude, and using vector geometry to solve geometric problems. 2. **Translation using a vector:** A translation moves every point of a figure or space by the same vector $\vec{v} = \langle v_x, v_y \rangle$. If a point $P = (x, y)$ is translated by $\vec{v}$, the new point $P' = (x + v_x, y + v_y)$. 3. **Vector addition and subtraction:** - Algebraically, if $\vec{a} = \langle a_x, a_y \rangle$ and $\vec{b} = \langle b_x, b_y \rangle$, then $$\vec{a} + \vec{b} = \langle a_x + b_x, a_y + b_y \rangle$$ $$\vec{a} - \vec{b} = \langle a_x - b_x, a_y - b_y \rangle$$ - Graphically, place the tail of $\vec{b}$ at the head of $\vec{a}$ for addition; the resultant vector is from the tail of $\vec{a}$ to the head of $\vec{b}$. For subtraction, add $\vec{a}$ to the negative of $\vec{b}$. 4. **Scalar multiplication:** Multiplying a vector $\vec{v} = \langle v_x, v_y \rangle$ by a scalar $k$ gives $$k\vec{v} = \langle kv_x, kv_y \rangle$$ Geometrically, this stretches or shrinks the vector by $|k|$ and reverses its direction if $k$ is negative. 5. **Magnitude of a vector:** The magnitude (length) of $\vec{v} = \langle v_x, v_y \rangle$ is $$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$$ This represents the distance from the origin to the point $(v_x, v_y)$. 6. **Using vector geometry to solve problems:** Vectors can represent points, directions, and displacements. For example, to find the midpoint $M$ of segment $AB$ with position vectors $\vec{a}$ and $\vec{b}$, $$\vec{m} = \frac{\vec{a} + \vec{b}}{2}$$ Vectors simplify calculations of distances, angles, and can solve geometric problems by algebraic manipulation. **Summary:** Vectors provide a powerful way to represent translations, combine movements, scale directions, measure lengths, and solve geometry problems efficiently.