1. **Stating the problem:** We want to understand the concept of vectors, their properties, and operations such as addition, subtraction, and scalar multiplication. 2. **Definition:** A vector is a quantity that has both magnitude and direction. It can be represented as $\vec{v} = \langle v_1, v_2, \ldots, v_n \rangle$ in $n$-dimensional space. 3. **Vector addition:** Given two vectors $\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$, their sum is $$\vec{a} + \vec{b} = \langle a_1 + b_1, a_2 + b_2 \rangle$$ This means we add corresponding components. 4. **Vector subtraction:** Similarly, $$\vec{a} - \vec{b} = \langle a_1 - b_1, a_2 - b_2 \rangle$$ 5. **Scalar multiplication:** Multiplying a vector by a scalar $k$ scales its magnitude: $$k \vec{a} = \langle k a_1, k a_2 \rangle$$ 6. **Magnitude of a vector:** The length or magnitude of $\vec{v} = \langle v_1, v_2 \rangle$ is $$|\vec{v}| = \sqrt{v_1^2 + v_2^2}$$ 7. **Dot product:** For vectors $\vec{a}$ and $\vec{b}$, $$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2$$ This is useful for finding angles and projections. 8. **Example:** Let $\vec{a} = \langle 3, 4 \rangle$ and $\vec{b} = \langle 1, 2 \rangle$. - Addition: $\vec{a} + \vec{b} = \langle 3+1, 4+2 \rangle = \langle 4, 6 \rangle$ - Subtraction: $\vec{a} - \vec{b} = \langle 3-1, 4-2 \rangle = \langle 2, 2 \rangle$ - Scalar multiplication (by 2): $2 \vec{a} = \langle 6, 8 \rangle$ - Magnitude of $\vec{a}$: $|\vec{a}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$ - Dot product: $\vec{a} \cdot \vec{b} = 3 \times 1 + 4 \times 2 = 3 + 8 = 11$ These operations form the foundation of vector algebra and are widely used in physics, engineering, and computer graphics.