1. Let's start by defining what vectors are. A vector is a quantity that has both magnitude and direction. 2. Vectors can be represented in component form as $\vec{v} = \langle v_x, v_y, v_z \rangle$ in 3D or $\vec{v} = \langle v_x, v_y \rangle$ in 2D. 3. To add two vectors $\vec{a} = \langle a_x, a_y \rangle$ and $\vec{b} = \langle b_x, b_y \rangle$, add their corresponding components: $\vec{a} + \vec{b} = \langle a_x + b_x, a_y + b_y \rangle$. 4. To find the magnitude (length) of a vector $\vec{v} = \langle v_x, v_y \rangle$, use the formula $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$. 5. The dot product of two vectors $\vec{a}$ and $\vec{b}$ is $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y$. 6. The dot product can also be expressed as $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$, where $\theta$ is the angle between the vectors. 7. The cross product of two vectors in 3D, $\vec{a} = \langle a_x, a_y, a_z \rangle$ and $\vec{b} = \langle b_x, b_y, b_z \rangle$, is $\vec{a} \times \vec{b} = \langle a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x \rangle$. 8. The cross product results in a vector perpendicular to both $\vec{a}$ and $\vec{b}$. 9. Vectors are fundamental in physics and engineering to represent forces, velocities, and other directional quantities. 10. If you have a specific vector problem or operation you want to solve, please provide the details!