Addition/subtraction and scalar multiplication | Year 11

01 / LEARN

Understand the idea.

Adding arrows head to tail

Add successive displacements by placing arrows head to tail. The resultant goes directly from the original start to the final endpoint. Algebraically, add horizontal changes together and vertical changes together; the route and its resultant have the same net displacement.

A scalar multiplies every component. For a non-zero vector, a positive scalar preserves direction and a negative scalar reverses it; the length is multiplied by the scalar's absolute value. Multiplying by zero gives the zero vector.

a u + b v = (a u_{x} + b v_{x}, a u_{y} + b v_{y})

Equal coordinate scales show u from the origin to three one, v to one three and u plus v to four four, with all three tails at the origin.

Subtraction is a displacement between arrowheads

Subtracting a vector means adding its opposite: $u - v = u + (- v)$ . If both arrows share a tail, the arrow from the head of $v$ to the head of $u$ represents this difference.

Apply an outside minus sign to every component or term inside its brackets. For example, $- (a, b) = (- a, - b)$ . Calculate scalar multiples first, then combine corresponding components, keeping brackets around negative numbers.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A linear combination

Given $u = (2, - 1)$ and $v = (- 3, 4)$ , find $3 u - 2 v$ .

Worked solution

1
Scale the vectors separately.
$\begin{aligned} 3 u & = 3 (2, - 1) = (6, - 3), \\ 2 v & = 2 (- 3, 4) = (- 6, 8) . \end{aligned}$
Scalar multiplication acts on both components.
2
Subtract component by component.
$3 u - 2 v = (6 - (- 6), - 3 - 8)$
Subtract the entire second vector, including its negative horizontal component.
3
Simplify the resultant.
$3 u - 2 v = (12, - 11)$
The horizontal change is positive twelve and the vertical change is negative eleven.

Answer

$(12, - 11)$ .

EXAMPLE 02

A route and a return journey

A canoe travels $(4, 1)$ km, then $(- 1, 3)$ km. Find the total displacement and the direct return displacement.

Worked solution

1
Add the two journey legs.
$d = (4, 1) + (- 1, 3) = (3, 4) km$
Successive horizontal and vertical changes accumulate independently.
2
Find the displacement that cancels the journey.
$r = - d = (- 3, - 4) km$
The return arrow points from the final position back to the original start.
3
Check the completed round trip.
$d + r = (3, 4) + (- 3, - 4) = (0, 0) km$
Returning to the starting position gives zero net displacement.

Answer

Total $(3, 4)$ km; direct return $(- 3, - 4)$ km.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find $2 (1, 3) - (4, - 2)$ .

Show answer & reasoning

$(- 2, 8)$ .

$2 (1, 3) = (2, 6)$ , then $(2, 6) - (4, - 2) = (2 - 4, 6 + 2) = (- 2, 8)$ . The vertical subtraction becomes addition because the component being subtracted is negative.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

For $a = (4, - 2)$ and $b = (- 1, 5)$ , find $a + b$ .
2 marks
For the same vectors, find $a - b$ .
2 marks
Find $- 3 (2, - 4) + 2 (- 1, 1)$ .
2 marks
Solve $2 x + (1, - 3) = (7, 5)$ .
2 marks
A drone flies $(3, 2)$ km, then $(- 5, 1)$ km, then $(1, - 4)$ km. Find its displacement and direct return vector.
3 marks
If $∣ u ∣ = 7$ , find $∣ - \frac{3}{2} u ∣$ and explain the direction.
3 marks

Read the worked solutions

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

For $a = (4, - 2)$ and $b = (- 1, 5)$ , find $a + b$ .
2 marks
Add corresponding components.
$(3, 3)$ .
For the same vectors, find $a - b$ .
2 marks
$4 - (- 1) = 5$ and $- 2 - 5 = - 7$ .
$(5, - 7)$ .
Find $- 3 (2, - 4) + 2 (- 1, 1)$ .
2 marks
The scaled vectors are $(- 6, 12)$ and $(- 2, 2)$ .
Add them.
$(- 8, 14)$ .
Solve $2 x + (1, - 3) = (7, 5)$ .
2 marks
$2 x = (6, 8)$ .
Divide both components by $2$ .
$x = (3, 4)$ .
A drone flies $(3, 2)$ km, then $(- 5, 1)$ km, then $(1, - 4)$ km. Find its displacement and direct return vector.
3 marks
Add all three legs componentwise.
Negate the resultant for the return.
Displacement $(- 1, - 1)$ km; return $(1, 1)$ km.
If $∣ u ∣ = 7$ , find $∣ - \frac{3}{2} u ∣$ and explain the direction.
3 marks
Multiply the length by $\frac{3}{2}$ .
The negative scalar reverses direction.
$10.5$ ; opposite to $u$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: vectors in the plane. Covers magnitude, components, geometry, scalar products, projections and proofs; follows the approved Specialist preparation structure.

Official curriculum source ↗

In this lesson

Understand the idea.

Adding arrows head to tail

Subtraction is a displacement between arrowheads

See the method in action.

A linear combination

Scale the vectors separately.

Subtract component by component.

Simplify the resultant.

A route and a return journey

Add the two journey legs.

Find the displacement that cancels the journey.

Check the completed round trip.

Try it yourself.

Check your understanding

Keep practising