Study Notes
Overview
Vectors are a fundamental concept in mathematics that describe quantities with both magnitude (size) and direction. For AQA GCSE, this topic bridges the gap between simple geometry and advanced algebra, making it a crucial area of study. At its core, a vector is a journey from one point to another. Understanding how to describe these journeys using column notation and how to combine them through addition and subtraction is the foundation for success. This topic is not just about calculation; it’s about logical reasoning and proof. Examiners frequently use vectors to test a candidate's ability to construct a logical argument, particularly in Higher Tier questions involving geometric proofs. Mastering vectors will not only prepare you for these specific questions but also strengthen your overall algebraic and geometric problem-solving skills, which are essential across the entire specification.
Key Concepts
Concept 1: Vector Notation and Representation
A vector is represented by a directed line segment. The arrow indicates the direction, and the length of the line represents the magnitude. In AQA exams, you will primarily use column vectors to represent these journeys. A column vector is written as two numbers stacked vertically within brackets, like so:
(3)
(4)
This vector represents a movement of 3 units to the right (positive x-direction) and 4 units up (positive y-direction). A negative number indicates movement in the opposite direction (left or down). It is critical to use brackets and not a fraction line, as using a fraction line will be marked as incorrect.
Concept 2: Vector Arithmetic: Addition, Subtraction, and Scalar Multiplication
Vector arithmetic follows specific rules. To add two vectors, you add their corresponding components. For example, if a = (2, 3) and b = (4, -1), then a + b = (2+4, 3-1) = (6, 2). Geometrically, this is like placing the vectors head-to-tail; the resultant vector goes from the start of the first vector to the end of the second.
Subtraction works similarly: you subtract the corresponding components. A key concept is that subtracting a vector is the same as adding its negative. The vector -b has the same magnitude as b but points in the opposite direction. So, a - b is the same as a + (-b).
Scalar multiplication involves multiplying a vector by a number (a scalar). This changes the vector's magnitude but not its direction. For example, 2a = 2 * (2, 3) = (4, 6). The vector is now twice as long but still points in the same direction.
Concept 3: Geometric Proofs with Vectors (Higher Tier)
This is where vectors become a powerful tool for proving geometric properties. The most common proofs involve showing that lines are parallel or that points are collinear (lie on the same straight line).
Two vectors are parallel if one is a scalar multiple of the other. For example, if you can show that vector AB = 2 * vector CD, then the line segment AB is parallel to CD and twice as long.
To prove that three points A, B, and C are collinear, you must show that the vector AB is a scalar multiple of the vector BC, and that they share a common point (B). This proves they lie on the same line.
Mathematical/Scientific Relationships
- Resultant Vector: The sum of two or more vectors, representing the overall displacement.
- Zero Vector: A vector with zero magnitude and no direction, represented as (0, 0).
- Position Vector: A vector that starts from the origin (0,0) and ends at a specific point.
- Parallel Vectors: Vector a is parallel to vector b if a = kb, where k is a non-zero scalar. (Must memorise)
- Collinear Points: Points A, B, and C are collinear if vector AB = k * vector BC for some scalar k. (Must memorise)
Practical Applications
Vectors are used extensively in physics to model forces, velocity, and acceleration. They are also fundamental in computer graphics for creating 3D models and animations, in aviation for navigation, and in engineering for analysing structures.