1. **Stating the problem:** We want to understand what vectors are and see an example with a drawing. 2. **Definition:** A vector is a quantity that has both magnitude (length) and direction. It is often represented as an arrow pointing from one point to another. 3. **Important rules:** - Vectors can be added or subtracted. - They can be multiplied by scalars (numbers). - The length of a vector $ \vec{v} = (x, y) $ is given by $$\|\vec{v}\| = \sqrt{x^2 + y^2}$$. 4. **Example:** Consider the vector $ \vec{v} = (3, 4) $. This means it moves 3 units in the x-direction and 4 units in the y-direction. 5. **Magnitude calculation:** $$\|\vec{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 6. **Explanation:** This vector points from the origin $(0,0)$ to the point $(3,4)$ in the plane, and its length is 5 units. 7. **Summary:** Vectors represent direction and length, useful in physics and math to describe movement, forces, and more.