Year 12 Maths Extension 1 (2027) Vectors

Vector representation & notation

Q: What does AB equal in position vectors?

AB equals b minus a, the head position minus the tail position.

Q: What is a position vector?

A vector with its tail at the origin, so OA equals a.

Q: What is the midpoint in vector form?

The midpoint of AB has position vector one half of a plus b.

20 practice questions 2 video lessons Theory + worked examples

Get to grips with vector representation and notation in NSW Year 12 Mathematics Extension 1. A vector has both magnitude and direction, pictured as a directed line segment and written in the bold, tilde or arrow notation used throughout the course.

You will learn to represent vectors geometrically, work with position vectors, recognise when two vectors are equal, and read magnitude and direction from a diagram — the foundational vector language behind every later Extension 1 vector topic.

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Practice questions

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Vectors - Vector Basics (by Class Mathematics) Watch
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Theory

A vector has magnitude and direction, drawn as a directed line segment. This NSW Year 12 Mathematics Extension 1 topic (NESA outcome ME1-12-02) covers vector notation, position vectors and the rule $\vec{A B} = b - a$ .

A vector has both magnitude and direction. It is drawn as a directed line segment (an arrow); the same vector may be shown by many parallel arrows of equal length.

Notation: $a$ in print, $\underset{\sim}{a}$ when handwritten, or $\vec{A B}$ for the vector from $A$ to $B$ ; its magnitude is $| a |$ or $| \vec{A B} |$ . A position vector has its tail at the origin, $\vec{O A} = a$ .

Two vectors are equal when they have the same magnitude and direction; $- a$ has the same magnitude but the opposite direction.

NESA link. Part of the Year 12 Introduction to vectors focus area, outcome ME1-12-02 ("operates with 2D and 3D vectors and uses 2D vectors to solve problems involving motion in two dimensions") with MAO-WM-01.

Tip-to-tail:

\vec{A B} + \vec{B C} = \vec{A C}

Position vectors give

\vec{A B} = b - a

(head minus tail).

\vec{A B} = b - a (head - tail), \vec{A B} = - \vec{B A} .

AB = b - a (head minus tail); AB = -BA

The midpoint of $A B$ has position vector $\frac{1}{2} (a + b)$ , and around any closed path the vectors sum to $0$ .

midpoint of AB has position vector one half (a + b)

Head minus tail. $\vec{A B} = b - a$ , not $a - b$ ; reversing the arrow negates the vector.

How to combine vectors

Tip to tail: to add, place the tail of the second at the head of the first; the sum runs from the first tail to the last head.
Head minus tail: with position vectors, $\vec{A B} = b - a$ .
Divide a segment: the point dividing $A B$ in ratio $m : n$ is $a + \frac{m}{m + n} (b - a)$ .
Closed paths return to the start, so their vectors sum to $0$ .

Example 1 — Triangle

In triangle

P Q R

\vec{P Q} = a

and

\vec{Q R} = b

. Find

\vec{P R}

and

\vec{R P}

Solution

$\vec{P R}$	$=$	$\vec{P Q} + \vec{Q R} = a + b$
$\vec{R P}$	$=$	$- (a + b)$

PR = a + b, RP = -a - b

$\vec{P R} = a + b$ , $\vec{R P} = - a - b$ .

Example 2 — Parallelogram diagonals

In parallelogram

O A B C

\vec{O A} = a

and

\vec{O C} = c

. Find

\vec{O B}

and

\vec{A C}

Solution

$\vec{O B}$	$=$	$a + c$
$\vec{A C}$	$=$	$c - a$

OB = a + c, AC = c - a

$\vec{O B} = a + c$ , $\vec{A C} = c - a$ .

Example 3 — Dividing a segment

A

B

have position vectors

a

b

. Find the midpoint

M

A B

, and

P

with

A P : P B = 1 : 2

Solution

$\vec{O M}$	$=$	$\frac{1}{2} (a + b)$
$\vec{O P}$	$=$	$a + \frac{1}{3} (b - a) = \frac{2}{3} a + \frac{1}{3} b$

M = (a+b)/2, P = (2a + b)/3

$M = \frac{1}{2} (a + b)$ , $P = \frac{2}{3} a + \frac{1}{3} b$ .

Example 4 — Closed path

In quadrilateral

A B C D

\vec{A B} = p

\vec{B C} = q

\vec{C D} = r

\vec{D A} = s

. Show

p + q + r + s = 0

Solution

Following the sides returns to the start:

p + q + r + s

=

\vec{A B} + \vec{B C} + \vec{C D} + \vec{D A} = \vec{A A} = 0

p + q + r + s = 0 around the closed path

So $p + q + r + s = 0$ .

Common pitfalls

Head minus tail.

\vec{A B} = b - a

, subtracting the tail from the head — not the other way round.

Direction of the arrow.

\vec{A B} = - \vec{B A}

: reversing the arrow negates the vector.

Equal vectors. Two vectors are equal if they match in magnitude and direction, wherever they are drawn.

Closed paths. Going right around a figure returns to the start, so the vectors sum to

0

Frequently asked questions

What is a vector?

A quantity with both magnitude and direction, drawn as a directed line segment.

What does AB equal in position vectors?

$\vec{A B} = b - a$ , the head position minus the tail position.

What is a position vector?

A vector with its tail at the origin, so $\vec{O A} = a$ .

When are two vectors equal?

When they have the same magnitude and the same direction, regardless of location.

What is the midpoint in vector form?

The midpoint of $A B$ has position vector $\frac{1}{2} (a + b)$ .

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2D vectors in component form

Vector representation & notation

📖 Theory

How to combine vectors

Common pitfalls

Frequently asked questions

Theory