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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