Ch 2 · 7.2–7.4 Vectors · O Level Mathematics 4024
Topic 2 of 2 in Transformations and Vectors — O Level Mathematics 4024 — 38 lessons.
Introduction
A vector is a movement with a size and a direction: "3 km east and 4 km north". Vectors let you prove geometric facts without measuring, such as showing that two lines are parallel or that three points lie on one straight line. (Syllabus sections 7.2–7.4.)
Key ideas — Notation and arithmetic
A vector can be written beginpmatrix 3 \\ -1 endpmatrix (3 right, 1 down), as overrightarrowAB, or as a bold letter a (underline it by hand). For a = beginpmatrix 3 \\ -1 endpmatrix and b = beginpmatrix -2 \\ 4 endpmatrix:
🧑🏫 Teacher's explanation
Think of a vector as a set of instructions for movement rather than a fixed point. If I tell you to 'walk 3 steps North and 2 steps East', that instruction is a vector. It doesn't matter where you start; the instruction itself is the same. In math, we represent this instruction using a column vector beginpmatrix x \ y endpmatrix, where x is the horizontal movement and y is the vertical movement.
📖 Definitions
🔍 Worked example — Standard
Find the magnitude of the vector v = beginpmatrix 5 \ 12 endpmatrix.
🔍 Worked example — Exam-style
In a triangle OAB, let vecOA = a and vecOB = b. Express overrightarrowAB in terms of a and b.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Magnitude
|beginpmatrix x \\ y endpmatrix | = √(x^2 + y^2)
🧑🏫 Teacher's explanation
Think of a vector as a set of instructions for a journey. If a vector beginpmatrix 3 \ 4 \endpmatrix tells you to walk 3 meters East and 4 meters North, the 'magnitude' is simply the total straight-line distance from your starting point to your destination. It is the 'as-the-crow-flies' distance, ignoring the specific direction.
📖 Definitions
🔍 Worked example — Easy
Find the magnitude of the vector v = beginpmatrix 5 \\ 12 \\endpmatrix.
🔍 Worked example — Standard
Given a = 3i - 4j, calculate |a|.
🔍 Worked example — Exam-style
A vector is defined by w = 2a - b, where a = beginpmatrix 4 \\ 3 \\endpmatrix and b = beginpmatrix 2 \\ -1 \\endpmatrix. Calculate |w|.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Position vectors and routes
The position vector of A is overrightarrowOA. To get from A to B, go via any route: overrightarrowAB = overrightarrowAO + overrightarrowOB = -a + b = b - a.
🧑🏫 Teacher's explanation
Think of a vector as a set of movement instructions rather than a fixed location. If you are standing at point A and want to get to point B, a vector overrightarrowAB tells you exactly how many steps to take East/West and how many North/South. It doesn't matter where you start; the instruction '3 steps East and 2 steps North' is always the same.
📖 Definitions
🔍 Worked example — Easy
Given position vectors a = beginpmatrix 3 \\ 2 endpmatrix and b = beginpmatrix 7 \\ 5 endpmatrix, find the vector overrightarrowAB and its magnitude |overrightarrowAB|.
🔍 Worked example — Standard
In a triangle OAB, overrightarrowOA = a and overrightarrowOB = b. M is the midpoint of AB. Express overrightarrowOM in terms of a and b.
🔍 Worked example — Exam-style
Points A, B, C are such that overrightarrowOA = a, overrightarrowOB = b, and overrightarrowOC = 3a + 2b. Show that A, B, C are collinear.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Parallel vectors and collinear points
Worked examples
OABC is a parallelogram with overrightarrowOA = a and overrightarrowOC = c. M is the midpoint of AB. overrightarrowOB = a + c; overrightarrowAB = c; overrightarrowOM = a + tfrac12c; overrightarrowCM = a - tfrac12c.
🔍 Worked example — Easy
Given a = beginpmatrix 3 \\ -2 endpmatrix and b = beginpmatrix 1 \\ 4 endpmatrix, find the vector c = 2a + b and its magnitude |c|.
🔍 Worked example — Standard
In a triangle OAB, overrightarrowOA = a and overrightarrowOB = b. P is a point on AB such that AP:PB = 2:1. Express overrightarrowOP in terms of a and b.
🔍 Worked example — Exam-style
Given overrightarrowPQ = kbeginpmatrix 2 \\ -3 endpmatrix and overrightarrowQR = beginpmatrix 4 \\ -6 endpmatrix. Find the value of k and show that P, Q, R are collinear.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) beginpmatrix 17 \\ 0 endpmatrix; p - q = beginpmatrix 6 \\ -1 endpmatrix, magnitude √(37) = 6.08. (2) beginpmatrix -6 \\ 8 endpmatrix; magnitude 10. (3) a + tfrac23(b - a) = tfrac13a + tfrac23b. (4) beginpmatrix 4 \\ -6 endpmatrix = -2beginpmatrix -2 \\ 3 endpmatrix, a scalar multiple, so parallel.