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 (NESA outcome ME1-12-02). 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