Use vectors in two dimensions and in three dimensions

Vectors are fundamental tools in mathematics and physics, allowing us to represent quantities that have both magnitude and direction. In two dimensions, vectors are typically expressed as column vectors (e.g., \begin{pmatrix} 3 \\ -2 \end{pmatrix}) or in terms of i and j components. In three dimensions, we extend this to include a k component, giving vectors like \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} or 3i + 2j + k. Understanding vectors is crucial for solving problems involving displacement, velocity, forces, and geometry in both 2D and 3D space.

This topic builds on GCSE work with column vectors and introduces new operations such as the scalar (dot) product, which is used to find angles between vectors and to determine perpendicularity. You will also learn to work with position vectors, vector equations of lines, and in 3D, vector equations of planes. These concepts are essential for further study in engineering, physics, and computer graphics, and they appear frequently in exam questions that require you to model real-world situations.

Mastering vectors requires a strong grasp of algebra and geometry. You will need to perform addition, subtraction, and scalar multiplication, as well as calculate magnitudes and unit vectors. In 3D, you will also encounter cross products (though this is more common in Further Maths). The key is to visualise vectors as directed line segments and to be comfortable with both component form and geometric interpretations.