Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form

Vectors are fundamental in A-Level Mathematics, allowing us to describe quantities that have both magnitude and direction, such as velocity, force, and displacement. This topic focuses on two key skills: calculating the magnitude (length) and direction (angle) of a vector given in component form, and converting between component form and magnitude-direction form. Mastering these conversions is essential for solving problems in mechanics, geometry, and further pure mathematics.

In component form, a vector is written as (x, y) or (x, y, z) in 2D or 3D, representing its horizontal and vertical components. Magnitude-direction form expresses the same vector using its length (magnitude) and an angle relative to a reference direction, typically the positive x-axis. The ability to switch between these forms is crucial for vector addition, resolving forces, and interpreting geometric transformations.

This topic builds on basic trigonometry and Pythagoras' theorem, and it underpins more advanced concepts such as scalar products, vector equations of lines, and parametric equations. By the end of this topic, you should be able to confidently calculate a vector's magnitude using Pythagoras, find its direction using inverse tangent (arctan), and reconstruct components from magnitude and direction using sine and cosine.