Vector equality and opposite vectors | Year 11

01 / LEARN

The idea, step by step

Equal arrows can start in different places

Two free vectors are equal when they have the same magnitude and direction. They need not start at the same point. In component form, equality means that the corresponding components are equal.

For directed segments, subtract start coordinates from end coordinates. Equal lengths alone are not enough to establish equal vectors: east and west arrows of length five are different vectors.

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

Reversing a directed segment

The opposite of a vector has the same magnitude and reverse direction. Negating both components reverses the arrow. In particular, reversing the endpoints of a segment negates its displacement.

The zero vector is the only vector equal to its own opposite. When comparing diagrams, read the arrowheads as carefully as the drawn lengths; a labelled line segment without an arrow may represent a length rather than a vector.

A common trap

Calling every pair of equal-length arrows equal vectors.

Instead: Check direction as well as length, or compare both components.

02 / WORKED EXAMPLES

See the thinking

Example 1

Equal vectors from coordinates

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

$\vec{A B} = (4 - 1, 6 - 2) = (3, 4)$ .
$\vec{C D} = (1 + 2, 5 - 1) = (3, 4)$ .
Their corresponding components agree.

\vec{A B} = \vec{C D}

, although the arrows start at different points.

Example 2

Finding an endpoint of an opposite arrow

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

$\vec{P Q} = (3, 4)$ , so its opposite is $(- 3, - 4)$ .
Add that displacement to the new starting point: $S = (4, 4) + (- 3, - 4)$ .

S = (1, 0)

Check your understanding

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

Show answer

$(2, - 5)$ .

Both components change sign when the direction reverses.

03 / WORKSHEET

Your turn

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

Worksheet PDF 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 ↗

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