1. The problem is to describe vectors and illustrate some examples. 2. A vector is a quantity that has both magnitude (length) and direction. 3. Vectors are often represented as arrows where the length corresponds to the magnitude and the arrow points in the direction. 4. Important properties of vectors include addition, scalar multiplication, and components along coordinate axes. 5. For example, a vector $ \vec{v} = \langle 3, 4 \rangle $ in 2D has components 3 along the x-axis and 4 along the y-axis. 6. The magnitude of $ \vec{v} $ is calculated by the formula: $$\|\vec{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 7. Vectors can be added by adding their components: $$\vec{a} = \langle a_x, a_y \rangle, \quad \vec{b} = \langle b_x, b_y \rangle$$ $$\vec{a} + \vec{b} = \langle a_x + b_x, a_y + b_y \rangle$$ 8. Scalar multiplication changes the magnitude but not the direction (unless scalar is negative). 9. Here are some example vectors drawn in the coordinate plane: $ \vec{v_1} = \langle 2, 3 \rangle $, $ \vec{v_2} = \langle -1, 4 \rangle $, and $ \vec{v_3} = \langle 0, -2 \rangle $.