Year 12 Maths Extension 1 (2027) Vectors

Operating with vectors in two dimensions

Q: How do you add vectors in component form?

Add matching components: i with i, j with j.

Q: How do you tell if two vectors are parallel?

They are parallel when a equals k times b, that is, their components are proportional.

Q: How do you find a perpendicular vector?

Swap the components and negate one, so a i plus b j becomes minus b i plus a j.

Q: What does a scalar multiple look like?

k a stretches a for positive k and reverses it for negative k.

20 practice questions 2 video lessons Theory + worked examples

Learn operating with vectors in two dimensions for NSW Year 12 Mathematics Extension 1. Vectors can be scaled, added and subtracted, combining geometric quantities such as displacements and velocities into a single resultant.

You will learn scalar multiplication, addition and subtraction using the triangle and parallelogram laws, how to test whether two vectors are parallel, and how to find a unit vector in a given direction — essential vector operations used across Extension 1.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 2 video(s)

Are The Two Vectors Parallel, Orthogonal, or Neither? Watch
Parallel and Perpendicular Vectors with Worked Examples Watch

Theory

Vectors add and subtract componentwise and scale by a scalar. Two vectors are parallel when $a = k b$ , and $a i + b j$ is perpendicular to $- b i + a j$ . This NSW Year 12 Mathematics Extension 1 topic is NESA outcome ME1-12-02.

Vectors are added or subtracted componentwise, and scaled by multiplying each component by a scalar. For $a = x_{1} i + y_{1} j$ , $b = x_{2} i + y_{2} j$ and scalar $k$ :

a \pm b = (x_{1} \pm x_{2}) i + (y_{1} \pm y_{2}) j, k a = k x_{1} i + k y_{1} j .

Two non-zero vectors are parallel when $a = k b$ for some scalar $k$ (proportional components). A vector perpendicular to $a i + b j$ is $- b i + a j$ — swap the components and negate one.

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.

Addition follows the triangle or parallelogram law.

k a

stretches

a

for

k > 0

and reverses it for

k < 0

a ∥ b ⟺ a = k b ⟺ \frac{x_{1}}{x_{2}} = \frac{y_{1}}{y_{2}}, a i + b j ⊥ - b i + a j .

parallel: a = k b (proportional components); perpendicular: swap and negate

Componentwise. Add $i$ with $i$ and $j$ with $j$ ; a scalar multiplies every component.

How to operate with vectors

Add or subtract matching components.
Scale by multiplying each component by the scalar.
Test parallel: check the components are proportional, or $a = k b$ .
Build a perpendicular: swap the components and negate one.

Example 1 — Linear combination

a = 3 i - j

and

b = 2 i + 4 j

, find

2 a + 3 b

Solution

$2 a$	$=$	$6 i - 2 j$
$3 b$	$=$	$6 i + 12 j$
$2 a + 3 b$	$=$	$12 i + 10 j$

2a + 3b = 12i + 10j

$2 a + 3 b = 12 i + 10 j$ .

Example 2 — Parallel

Find

k

so that

6 i + k j

is parallel to

3 i - 2 j

Solution

Components must be proportional.

\frac{6}{3}

=

\frac{k}{- 2} \Rightarrow 2 = \frac{k}{- 2} \Rightarrow k = - 4

k = -4

$k = - 4$ .

Example 3 — Perpendicular

Find a vector perpendicular to

5 i + 2 j

Solution

Swap the components and negate one.

5 i + 2 j

\to

- 2 i + 5 j

perpendicular: -2i + 5j (or 2i - 5j)

For example, $- 2 i + 5 j$ (or $2 i - 5 j$ ).

Example 4 — Subtract and test

Let

a = 2 i + 3 j

b = i - 4 j

. Find

a - b

and state whether

a ∥ b

Solution

a - b

=

(2 - 1) i + (3 + 4) j = i + 7 j

Parallel? $\frac{2}{1} \neq \frac{3}{- 4}$ , so no.

a - b = i + 7j; not parallel

$a - b = i + 7 j$ ; not parallel.

Common pitfalls

Mixing components. Add

i

with

i

and

j

with

j

, never across.

Parallel test. Parallel means proportional components

(\frac{x_{1}}{x_{2}} = \frac{y_{1}}{y_{2}})

, or

a = k b

Perpendicular sign. Swap the components and negate exactly one:

a i + b j \to - b i + a j

Sign of the scalar. A negative scalar reverses the direction as well as scaling the length.

Frequently asked questions

How do you add vectors in component form?

Add matching components: $i$ with $i$ , $j$ with $j$ .

How do you multiply a vector by a scalar?

Multiply every component by the scalar; a negative scalar reverses direction.

How do you tell if two vectors are parallel?

They are parallel when $a = k b$ , i.e. their components are proportional.

How do you find a perpendicular vector?

Swap the components and negate one: $a i + b j \to - b i + a j$ .

What does a scalar multiple look like?

$k a$ stretches $a$ for $k > 0$ and reverses it for $k < 0$ .

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2D vectors in component form

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3D vectors (component form, magnitude, operations)

Operating with vectors in two dimensions

📖 Theory

How to operate with vectors

Common pitfalls

Frequently asked questions

Theory