Vector interpretation of addition and conjugation | Year 11

01 / LEARN

The idea, step by step

Addition is a two-component translation

Writing $z = a + b i$ as vector $(a, b)$ makes complex addition identical to component vector addition. The point $z + w$ can be found by placing the displacement for $w$ at the end of the vector for $z$ .

Subtracting $w$ means adding the opposite displacement. The vector from point $A$ to point $B$ is always $B - A$ , not $A - B$ .

Conjugation and real scalar multiplication

Conjugation reflects in the real axis, so it preserves lengths and horizontal coordinates. Applying it twice returns the original number: $\bar{\bar{z}} = z$ .

Multiplication by a real $k$ scales both components. A positive $k$ preserves direction; a negative $k$ also reverses direction. It scales distances from the origin by $∣ k ∣$ . These facts let you check component calculations geometrically.

A common trap

Calling conjugation a reflection in the imaginary axis.

Instead: Conjugation leaves the real coordinate unchanged and reverses the imaginary coordinate, so the mirror is the real axis.

02 / WORKED EXAMPLES

See the thinking

Example 1

Add using two representations

Let $u = 2 + i$ and $v = - 1 + 3 i$ . Find $u + v$ and describe its construction.

Add components: $(2 - 1) + (1 + 3) i = 1 + 4 i$ .
Start at $(2, 1)$ and move $(- 1, 3)$ to reach $(1, 4)$ .
The origin and this endpoint form the resultant vector.

u + v = 1 + 4 i

Example 2

Find a reflected midpoint

Let $z = - 3 + 2 i$ . Find the midpoint of the points $z$ and $\bar{z}$ .

The reflected point is $\bar{z} = - 3 - 2 i$ .
The midpoint is $(z + \bar{z}) / 2 = (- 6) / 2 = - 3$ .
The midpoint lies on the mirror axis, the real axis.

- 3

Check your understanding

What translation maps $z$ to $z + 3 - 2 i$ ?

Show answer

Three units right and two units down.

The added complex number is displacement $(3, - 2)$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $u + v$ and $u - v$ for $u = 3 - 2 i$ and $v = - 1 + i$ .
3 marks
Find the vector from $A = - 2 + i$ to $B = 1 - 3 i$ .
2 marks
Find the midpoint between $2 + 5 i$ and $- 4 - i$ .
3 marks
Describe the effect of multiplying all complex numbers by $- 2$ .
3 marks
Prove $z + \bar{z}$ is real and $z - \bar{z}$ has real part zero for every complex $z$ .
4 marks
A parallelogram has successive vertices $0$ , $z$ , $z + w$ , $w$ . For $z = 4 + i$ and $w = - 1 + 2 i$ , find its centre.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $u + v$ and $u - v$ for $u = 3 - 2 i$ and $v = - 1 + i$ .
3 marks
Add or subtract the two real components and the two imaginary components.
$u + v = 2 - i$ ; $u - v = 4 - 3 i$ .
Find the vector from $A = - 2 + i$ to $B = 1 - 3 i$ .
2 marks
$B - A = (1 - 3 i) - (- 2 + i) = 3 - 4 i$ .
$3 - 4 i$ .
Find the midpoint between $2 + 5 i$ and $- 4 - i$ .
3 marks
Average components: $(2 - 4) / 2 = - 1$ and $(5 - 1) / 2 = 2$ .
$- 1 + 2 i$ .
Describe the effect of multiplying all complex numbers by $- 2$ .
3 marks
Both coordinates double in magnitude and reverse sign.
A dilation by factor $2$ about the origin followed by a half-turn.
Prove $z + \bar{z}$ is real and $z - \bar{z}$ has real part zero for every complex $z$ .
4 marks
Write the conjugate as $a - b i$ .
Add and subtract componentwise; each result has the required zero component.
If $z = a + b i$ , the expressions are $2 a$ and $2 b i$ . The second is pure imaginary when $b \neq 0$ , and is zero when $b = 0$ .
A parallelogram has successive vertices $0$ , $z$ , $z + w$ , $w$ . For $z = 4 + i$ and $w = - 1 + 2 i$ , find its centre.
3 marks
The opposite vertices $0$ and $z + w = 3 + 3 i$ have midpoint $(3 + 3 i) / 2$ .
Diagonals of a parallelogram share their midpoint.
$\frac{3}{2} + \frac{3}{2} i$ .

Checking saved progress…

Continue learning

Linear factorisation over the complex numbers

Next lesson

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, real and complex numbers: Cartesian arithmetic, conjugates, quadratic roots and Argand geometry. Polar form and roots of unity are developed in Year 12.

Official curriculum source ↗

Back to Complex numbers