Vectors

Overview

Vectors are a cornerstone of A-Level Mathematics, providing a powerful language to describe quantities that possess both magnitude and direction. This topic, specification point 3.12 for Edexcel, moves beyond simple GCSE concepts into a rigorous exploration of 2D and 3D space. A firm grasp of vectors is essential not only for pure mathematics, where they are used for geometric proofs involving lines and shapes, but also for mechanics, where they are fundamental to modelling forces, velocity, and acceleration. Examiners frequently test vector reasoning in multi-step problems that require candidates to construct logical arguments. Typical questions range from calculating magnitudes and distances to proving properties like collinearity or finding the point of intersection of two lines. Success in this area demonstrates a deep understanding of spatial relationships and mathematical structure.

Key Concepts

Concept 1: Vector Notation and Representation

A vector can be represented in several ways, and candidates must be fluent in all of them. The most common are column vectors and i, j, k notation. A 2D displacement vector that moves 3 units right (positive x-direction) and 4 units up (positive y-direction) can be written as a column vector (3, 4) or as 3i + 4j. In 3D, we add a third component, k, for the z-direction. For example, 2i - 5j + k corresponds to the column vector (2, -5, 1). It is crucial to distinguish between position vectors, which give the coordinates of a point relative to the origin (O), and displacement vectors, which describe the journey between two points. The displacement vector from point A to point B, denoted AB, is found by subtracting the position vector of the start point from the end point: AB = OB - OA = b - a. Confusing this with a + b is a frequent source of lost marks.

Concept 2: Magnitude and Unit Vectors

The magnitude (or modulus) of a vector is its length, a scalar quantity. It is calculated using Pythagoras's theorem in 2D or 3D. For a vector v = xi + yj + zk, its magnitude |v| is given by the formula |v| = sqrt(x^2 + y^2 + z^2). A common error is mishandling negative components; remember that squaring a negative number yields a positive result (e.g., (-4)^2 = 16). A unit vector is a vector with a magnitude of 1. To find the unit vector in the direction of a vector v, you divide the vector by its magnitude: û = v / |v|. This is a key skill for questions involving directions in space.

Concept 3: Vector Arithmetic

Vectors can be added, subtracted, and multiplied by scalars. To add or subtract vectors, you simply perform the operation on the corresponding components. For example, if a = (a1, a2) and b = (b1, b2), then a + b = (a1 + b1, a2 + b2). Geometrically, this is represented by the triangle law (placing vectors head-to-tail) or the parallelogram law. Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative, in which case the direction is reversed). If v = (x, y), then kv = (kx, ky). This concept is the foundation for understanding parallel vectors: two vectors a and b are parallel if and only if a = kb for some non-zero scalar k.

Concept 4: Geometric Proofs with Vectors

Vectors are a powerful tool for proving geometric properties. A classic exam question involves proving that three points, A, B, and C, are collinear (lie on the same straight line). To earn full marks, a candidate must provide a two-part conclusion: 1. Show that the vectors AB and BC are parallel (i.e., AB = k * BC). 2. State that they share a common point (point B). Simply proving they are parallel is insufficient, as parallel vectors can exist anywhere in space. Other common proofs involve showing that a shape is a specific type of quadrilateral (e.g., a parallelogram, where you must show that opposite sides are equal and parallel, AB = DC).

Mathematical Relationships

• Displacement Vector: AB = b - a (Must memorise)
• Magnitude of a 3D Vector: |v| = sqrt(x^2 + y^2 + z^2) (Given on formula sheet)
• Unit Vector: û = v / |v| (Must memorise)
• Scalar (Dot) Product: a · b = |a||b|cos(θ) (Given on formula sheet). This is used to find the angle between two vectors.
• Condition for Parallel Vectors: a = kb for some scalar k. (Must memorise)
• Condition for Perpendicular Vectors: a · b = 0. (Must memorise)

Practical Applications

Vectors are not just an abstract concept; they are essential in many fields. In Physics and Engineering, vectors are used to model forces, velocity, acceleration, and momentum. For example, when calculating the resultant force on an object, engineers add the force vectors acting on it. In Computer Graphics, vectors are used to determine the position and orientation of objects in a 3D virtual world, as well as for lighting calculations (using the dot product to determine how light reflects off a surface). In Aviation and Marine Navigation, pilots and captains use vectors to calculate their course, factoring in the velocity of their vehicle and the velocity of the wind or current.