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06062025–2027 syllabus

ADDITIONAL MATHEMATICS · CHAPTER 13

Vectors in two dimensions

Represent direction and magnitude consistently, then use components to prove geometry and model motion.

Additional Mathematics4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • use column, directed-segment, position and i–j notation
  • calculate magnitudes and unit vectors
  • solve vector-geometry problems
  • compose velocities and determine collision conditions

AT A GLANCE

Syllabus0606Coverage2025–2027Sections4LevelAdditional Mathematics

INTRODUCTION · THE BIG IDEA

Represent direction and magnitude consistently, then use components to prove geometry and model motion.

A vector describes a change: it has magnitude and direction but no fixed location. Component form turns geometry and motion into simultaneous algebra.

Position vectors locate points from a common origin. Differences of position vectors give directed displacements, so order and notation matter.

01

SECTION 01

Notation and operations

Core concept

Add and subtract corresponding components and multiply every component by a scalar. Equal vectors have equal matching components.

RULE 1
(a,b) + (c,d) = (a+c,b+d)
RULE 2
AB⃗ = OB⃗ − OA⃗
RULE 3
|(a,b)| = √(a²+b²)
Accepted vector forms
FormExample
column(a over b)
directed segmentAB⃗
bold or underlined letterp
unit-vector formai − bj
02

SECTION 02

Position and unit vectors

Core concept

A position vector gives a point's displacement from the origin. Divide a non-zero vector by its magnitude to obtain a unit vector in the same direction.

RULE 1
unit vector along a = a/|a|
RULE 2
OP⃗ = position vector of P
Original worked example

Create a unit direction

  1. Let a = 6i − 8j.
  2. Find |a| = √(36 + 64) = 10.
  3. Divide each component by 10.

Answer: Unit vector = (3/5)i − (4/5)j.

03

SECTION 03

Vector geometry

Core concept

Build routes between points using head-to-tail addition. Parallel vectors are scalar multiples; equal-direction ratios can establish collinearity or division of a segment.

DETAILED EXPLANATION

  • A point dividing AB internally can be written as A plus a fraction of AB⃗.
  • If AP⃗ = kAB⃗, then A, P and B are collinear.
  • For P between A and B, the scalar k lies between 0 and 1.
RULE 1
OM⃗ = ½(OA⃗ + OB⃗)
RULE 2
AP⃗ = OP⃗ − OA⃗
Triangle with midpoint vector constructionOABMabAM = 1/2(b − a)
If M is the midpoint of OB, then OM = 1/2b and AM = 1/2b − a; scalar multiples establish parallel directions.
04

SECTION 04

Velocity vectors and collisions

Core concept

Compose velocities by vector addition. For constant velocity v from initial position r₀, position is r(t) = r₀ + tv. A collision requires both particles to have identical position components at one common time.

RULE 1
resultant velocity = v₁ + v₂ + …
RULE 2
r(t) = r₀ + tv
ORIGINAL STUDY DIAGRAMTest for a collision
1Write each position vector
2Equate x-components
3Equate y-components
4Solve for one common time
5Confirm the shared position

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • Vector operations act component by component.
  • A displacement is final position minus initial position.
  • A unit vector is a vector divided by its magnitude.
  • Scalar multiples prove parallel directions and collinearity.
  • Collision requires equal positions at the same time.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • use column, directed-segment, position and i–j notation
  • calculate magnitudes and unit vectors
  • solve vector-geometry problems
  • compose velocities and determine collision conditions
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