Vector equality and opposite vectors | Year 11

01 / LEARN

Understand the idea.

Equal arrows can start in different places

Think of a free vector as an instruction for moving: the same horizontal change and the same vertical change give the same vector wherever you start. Thus equal vectors have matching components, as well as equal magnitude and direction.

To obtain $\vec{A B}$ , subtract the coordinates of $A$ from those of $B$ . The order follows the arrow: end minus start. Comparing the two component changes is more reliable than estimating arrows from a sketch.

\vec{A B} = (x_{B} - x_{A}, y_{B} - y_{A})

Equal coordinate scales show u from the origin to four one and negative u divided by two from the origin to minus two minus one half, with opposite direction and half the length.

Reversing a directed segment

The opposite vector undoes the original movement. Negate every component, so travelling along $u$ and then $- u$ returns to the start. Reversing the endpoints gives $\vec{B A} = - \vec{A B}$ .

Negating a vector preserves its length but reverses its direction. Equal length alone therefore does not establish vector equality. Only $0$ equals its own opposite, since $v = - v$ implies $2 v = 0$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Equal vectors from coordinates

Let $A = (1, 2)$ , $B = (4, 6)$ , $C = (- 2, 1)$ and $D = (1, 5)$ . Compare $\vec{A B}$ and $\vec{C D}$ .

Worked solution

1
Calculate the first displacement.
$\vec{A B} = (4 - 1, 6 - 2) = (3, 4)$
Subtract the starting coordinates from the ending coordinates.
2
Calculate the second displacement.
$\vec{C D} = (1 - (- 2), 5 - 1) = (3, 4)$
Subtracting a negative coordinate adds its positive value.
3
Compare both components.
$\vec{A B} = \vec{C D}$
Both arrows move three units right and four units up, even though they start at different points.

Answer

$\vec{A B} = \vec{C D}$ , although the arrows start at different points.

EXAMPLE 02

Finding an endpoint of an opposite arrow

Let $P = (2, - 1)$ , $Q = (5, 3)$ and $R = (4, 4)$ . Find $S$ if $\vec{R S} = - \vec{P Q}$ .

Worked solution

1
Find the original displacement.
$\vec{P Q} = (5 - 2, 3 - (- 1)) = (3, 4)$
The arrow runs from P to Q.
2
Reverse the displacement.
$\vec{R S} = - \vec{P Q} = (- 3, - 4)$
An opposite vector reverses both component changes.
3
Add the change to the new start.
$S = (4, 4) + (- 3, - 4) = (1, 0)$
An endpoint equals its starting point plus the required displacement.

Answer

$S = (1, 0)$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

If $u = (- 2, 5)$ , what is $- u$ ?

Show answer & reasoning

$(2, - 5)$ .

Negate both components: $- (- 2, 5) = (2, - 5)$ . Changing only one sign would reflect the arrow rather than reverse it.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find the opposite of $(7, - 3)$ .
2 marks
Are $(3, 4)$ and $(4, 3)$ equal vectors? Explain.
2 marks
If $(a - 1, 2 b) = (4, - 6)$ , find $a$ and $b$ .
2 marks
Let $A = (- 1, 3)$ and $B = (4, 1)$ . Find both $\vec{A B}$ and $\vec{B A}$ .
3 marks
A vector equal to $(2, - 5)$ starts at $(6, 1)$ . Find its endpoint.
2 marks
Solve $v = - v$ and explain the result.
3 marks

Read the worked solutions

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

Find the opposite of $(7, - 3)$ .
2 marks
Negate each component.
$(- 7, 3)$ .
Are $(3, 4)$ and $(4, 3)$ equal vectors? Explain.
2 marks
Their components differ, even though both magnitudes are $5$ .
No.
If $(a - 1, 2 b) = (4, - 6)$ , find $a$ and $b$ .
2 marks
Equate components: $a - 1 = 4$ and $2 b = - 6$ .
$a = 5$ , $b = - 3$ .
Let $A = (- 1, 3)$ and $B = (4, 1)$ . Find both $\vec{A B}$ and $\vec{B A}$ .
3 marks
Subtract the start from the end for each arrow.
The answers are opposites.
$(5, - 2)$ and $(- 5, 2)$ .
A vector equal to $(2, - 5)$ starts at $(6, 1)$ . Find its endpoint.
2 marks
Add the component changes to the starting coordinates.
$(8, - 4)$ .
Solve $v = - v$ and explain the result.
3 marks
Adding $v$ to both sides gives $2 v = 0$ .
Only zero has no distinction between a direction and its reverse.
$v = 0$ .

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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.

Equal arrows can start in different places

Reversing a directed segment

See the method in action.

Equal vectors from coordinates

Calculate the first displacement.

Calculate the second displacement.

Compare both components.

Finding an endpoint of an opposite arrow

Find the original displacement.

Reverse the displacement.

Add the change to the new start.

Try it yourself.

Check your understanding

Keep practising