Transformations and vectors
Track points through transformations and describe positions using directed vectors.
What you will learn
- Apply transformations in the stated order.
- Divide a line segment using a ratio.
Order changes the result
Reflecting a point in y=x swaps its coordinates. Rotating 90° anticlockwise about the origin maps (x,y) to (−y,x). Apply the first transformation to the original point and the second to the first image. Reversing the order generally produces a different result.
Always include the centre of rotation and the direction. Coordinate rules about the origin cannot be used unchanged for a different centre.
Vectors represent displacement
If OA=a and OB=b, then AB=b−a. To reach a point partway from A to B, start at a and add the required fraction of b−a. For AP:PB=m:n, that fraction is m/(m+n), not m/n.
The resulting coefficients of a and b add to one for any point on the line through A and B. For internal division, both coefficients lie between zero and one.
Worked example
OA=a and OB=b. P divides AB in the ratio AP:PB=1:3. Express OP.
Show the worked solution
- P is one quarter of the way from A to B.
- Start at A: OP=a+(1/4)(b−a).
- Collect coefficients of a and b.
Answer OP=(3/4)a+(1/4)b
Common mistakes
- AB is b−a, not a−b.
- Do not rotate the original point twice when the question describes two different transformations.
What can you explain now?
Reflect (3,−2) in y=x, then rotate the image 90° anticlockwise about the origin.
Compare with the explanation
(−3,−2)
The reflection gives (−2,3). Applying (x,y)→(−y,x) gives (−3,−2).
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Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources