Addition/subtraction and scalar multiplication | Year 11

01 / LEARN

The idea, step by step

Adding arrows head to tail

Successive displacements add. Place the tail of the second arrow at the head of the first; the resultant runs from the first starting point to the final endpoint. Components add because horizontal and vertical changes accumulate independently.

Multiplying a vector by a positive scalar stretches or shrinks it without changing direction. A negative scalar also reverses its direction. Magnitude therefore scales by the absolute value of the scalar.

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

Subtraction is a displacement between arrowheads

To calculate $u - v$ , add the opposite of $v$ . When both arrows start at the origin, the vector from the head of $v$ to the head of $u$ is $u - v$ .

Brackets matter. A minus sign before a vector sum reverses every term inside the brackets. For vector equations, ordinary collection of like terms works because addition and scalar multiplication obey the usual distributive laws.

A common trap

Writing $∣ k u ∣ = k ∣ u ∣$ when $k$ is negative.

Instead: Lengths cannot be negative: use $∣ k u ∣ = ∣ k ∣ ∣ u ∣$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A linear combination

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

$3 u = (6, - 3)$ and $2 v = (- 6, 8)$ .
Subtract corresponding components: $(6 - (- 6), - 3 - 8)$ .

(12, - 11)

Example 2

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.

Add the legs: $(4, 1) + (- 1, 3) = (3, 4)$ .
The direct return must cancel the total displacement, so reverse both components.

Total

(3, 4)

km; direct return

(- 3, - 4)

km.

Check your understanding

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

Show answer

$(- 2, 8)$ .

Double first, then subtract: $(2, 6) - (4, - 2) = (- 2, 8)$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF 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 $3 / 2$ .
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 ↗

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