Perpendicularity | Year 11

01 / LEARN

The idea, step by step

An algebraic right-angle test

Two non-zero vectors are perpendicular exactly when their dot product is zero. This follows from the geometric formula because the cosine of 90 degrees is zero.

For a non-zero vector $(a, b)$ , the vectors $(- b, a)$ and $(b, - a)$ are perpendicular to it. They result from quarter-turn rotations in opposite senses. They are useful when one component is zero and a gradient approach would be awkward.

(a, b) \cdot (- b, a) = 0

Use the angle's actual vertex

To test whether angle ABC is a right angle, use BA and BC. Both vectors must start at B, the vertex. Although reversing a vector preserves a zero dot product, choosing the wrong pair of sides can test a completely different angle.

A zero vector also has dot product zero with every vector, but it cannot form an angle. Check that the geometric vectors are non-zero before concluding that a right angle exists.

A common trap

Using the negative reciprocal gradient rule when a direction is vertical or horizontal, without treating those cases.

Instead: A dot-product test handles all non-zero component vectors uniformly.

02 / WORKED EXAMPLES

See the thinking

Example 1

Finding an unknown component

Find k if $(3, k)$ is perpendicular to $(4, - 2)$ .

Set their dot product equal to zero: $12 - 2 k = 0$ .
Solve to get $k = 6$ .

k = 6

Example 2

A right angle in a triangle

A=(1,1), B=(4,2), C=(3,5). Show that angle ABC is a right angle.

BA= $(- 3, - 1)$ and BC= $(- 1, 3)$ .
Their dot product is $3 - 3 = 0$ .
Both are non-zero vectors from B.

∠ A B C = 90^{\circ}

Check your understanding

Give a non-zero vector perpendicular to $(5, - 2)$ .

Show answer

For example $(2, 5)$ .

$5 (2) + (- 2) (5) = 0$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Are $(2, 3)$ and $(6, - 4)$ perpendicular?
2 marks
Find k if $(k, 5)$ is perpendicular to $(2, - 6)$ .
2 marks
Find a unit vector perpendicular to $(3, 4)$ .
2 marks
A=(0,0), B=(2,1), C=(1,3). Decide whether angle ABC is a right angle.
3 marks
For u=(t,2), v=(3,t), find all real t making u perpendicular to v.
3 marks
Explain why $0 \cdot (1, 2) = 0$ does not show a right angle.
2 marks

Read the worked solutions

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

Are $(2, 3)$ and $(6, - 4)$ perpendicular?
2 marks
Their dot product is $12 - 12 = 0$ .
Yes.
Find k if $(k, 5)$ is perpendicular to $(2, - 6)$ .
2 marks
$2 k - 30 = 0$ .
$k = 15$ .
Find a unit vector perpendicular to $(3, 4)$ .
2 marks
Rotate to $(- 4, 3)$ and divide by its length five.
For example $(- 4 / 5, 3 / 5)$ .
A=(0,0), B=(2,1), C=(1,3). Decide whether angle ABC is a right angle.
3 marks
BA= $(- 2, - 1)$ ; BC= $(- 1, 2)$ .
Their dot product is zero.
Yes.
For u=(t,2), v=(3,t), find all real t making u perpendicular to v.
3 marks
$u \cdot v = 3 t + 2 t = 5 t$ .
Set it equal to zero; the resulting vectors are both non-zero.
$t = 0$ .
Explain why $0 \cdot (1, 2) = 0$ does not show a right angle.
2 marks
The geometric perpendicularity conclusion requires both vectors to be non-zero.
The zero vector has no direction, so it forms no defined angle.

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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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