Projections onto coordinate axes | Year 11

01 / LEARN

The idea, step by step

Keep one component, discard the other

Orthogonal projection onto the x-axis sends (x,y) to (x,0). Projection onto the y-axis sends it to (0,y). The discarded movement is perpendicular to the target axis.

Projection is not reflection. Reflection reverses the perpendicular component, while projection removes it.

Projection is idempotent

The x-axis matrix is $P_{x} = (\begin{array}{cccccccccc} 1 & 0 \\ 0 & 0 \end{array})$ . It satisfies $P_{x}^{2} = P_{x}$ : projecting a point already on the axis does nothing more.

Coordinate projections are singular. Different points with the same retained component have the same image, so an inverse cannot recover the removed component.

A common trap

Using a negative entry instead of zero in a projection matrix.

Instead: Zero removes a component; a negative entry instead reverses or scales it.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two coordinate projections

Project $(3, - 4)$ onto the x-axis and $(- 2, 5)$ onto the y-axis.

The first keeps x=3 and removes y.
The second removes x and keeps y=5.

(3, 0)

and

(0, 5)

Example 2

Combine projections

Find $P_{x} + P_{y}$ and $P_{x} P_{y}$ .

The two projection matrices add to the identity.
Projecting first onto y, then x, removes both components.

P_{x} + P_{y} = I

and

P_{x} P_{y} = 0

Check your understanding

Find $P_{y} {(7, - 2)}^{T}$ .

Show answer

${(0, - 2)}^{T}$ .

Only the y-coordinate remains.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Give the matrix for projection onto the y-axis.
2 marks
Find $P_{x} {(- 5, 6)}^{T}$ .
2 marks
Find all points whose x-axis projection is $(2, 0)$ .
2 marks
Calculate $P_{y}^{2}$ .
2 marks
Explain why a coordinate projection has no inverse.
2 marks
Compare the x-axis projection and x-axis reflection of $(3, 4)$ .
2 marks

Read the worked solutions

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

Give the matrix for projection onto the y-axis.
2 marks
The unit-vector images are (0,0) and (0,1).
$(\begin{array}{cccccccccc} 0 & 0 \\ 0 & 1 \end{array})$ .
Find $P_{x} {(- 5, 6)}^{T}$ .
2 marks
Keep x and remove y.
${(- 5, 0)}^{T}$ .
Find all points whose x-axis projection is $(2, 0)$ .
2 marks
The x-coordinate must be 2; the y-coordinate is unrestricted.
$(2, t)$ for any real t.
Calculate $P_{y}^{2}$ .
2 marks
Multiply the two diagonal projection matrices.
The surviving diagonal entry remains 1.
$P_{y}$ .
Explain why a coordinate projection has no inverse.
2 marks
Many inputs have the same image, and the determinant is zero.
It loses one coordinate.
Compare the x-axis projection and x-axis reflection of $(3, 4)$ .
2 marks
Projection discards y; reflection reverses y.
Projection $(3, 0)$ ; reflection $(3, - 4)$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

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