1. Let's start by understanding what a vector is. A vector is a quantity that has both magnitude (length) and direction. 2. Vectors are often represented as arrows in space, where the length of the arrow corresponds to the magnitude and the arrow points in the direction. 3. The most common way to represent a vector algebraically is by its components, for example, in 2D: $\vec{v} = \langle v_x, v_y \rangle$. 4. Important operations with vectors include addition, subtraction, scalar multiplication, dot product, and cross product (in 3D). 5. Vector addition is done component-wise: if $\vec{a} = \langle a_x, a_y \rangle$ and $\vec{b} = \langle b_x, b_y \rangle$, then $\vec{a} + \vec{b} = \langle a_x + b_x, a_y + b_y \rangle$. 6. The magnitude of a vector $\vec{v} = \langle v_x, v_y \rangle$ is given by $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$. 7. The dot product of two vectors $\vec{a}$ and $\vec{b}$ is $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y$ and it relates to the angle $\theta$ between them by $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$. 8. If you have a specific question about vectors, such as finding the resultant vector, magnitude, direction, or projection, please ask!