I’m revising transformations to strengthen my fundamentals, and I’m stuck on composing them in the right order.
Example: Take triangle PQR with P(2, -3), Q(4, 1), and R(0, 0). The instruction is: reflect the triangle in the line y = x, then rotate it 90° counterclockwise about the origin. I keep second-guessing the formulas and the order.
My (probably wrong) attempt: I reflected using (x, y) -> (x + y, x – y), and then I rotated using (x, y) -> (x, y) because I thought a 90° turn would bring it back if one coordinate was zero. That clearly doesn’t make sense once I plot it, so I think I’ve messed up both steps.
Could someone explain the clean, step-by-step way to apply these two transformations to the coordinates without drawing everything? Also, if I swap the order (rotate first, then reflect in y = x), do I end up at the same image, or does the order matter here?
Side question: Is there a quick, reliable method (like a small 2×2 matrix approach) that I can use to combine these without re-deriving rules every time, or is that overkill for this level?
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3 Responses
Love this question! Think of “reflect in y = x” as swapping name tags: every point (x, y) just trades places to become (y, x). Then a 90° counterclockwise rotation is the quarter-turn rule (x, y) → (−y, x). So do it step by step:
– Reflect first: P(2, −3) → (−3, 2), Q(4, 1) → (1, 4), R(0, 0) → (0, 0).
– Then rotate: (−3, 2) → (−2, −3), (1, 4) → (−4, 1), (0, 0) → (0, 0).
Final positions: P′′(−2, −3), Q′′(−4, 1), R′′(0, 0). A neat surprise: this composition is exactly the reflection across the y-axis, (x, y) → (−x, y). Your earlier rule (x + y, x − y) isn’t a reflection; it distorts lengths, so it couldn’t match “mirror + turn.”
If you like quick, reliable methods, 2×2 matrices are your best pocket tool. Reflection in y = x is the swap matrix [[0, 1], [1, 0]], and 90° CCW rotation is [[0, −1], [1, 0]]. Multiply in the order you perform them: rotation × reflection = [[−1, 0], [0, 1]] (reflect in the y-axis). If you swap the order (reflect after rotating), you get [[1, 0], [0, −1]] (reflect in the x-axis). So yes, the order matters-just like putting on socks before shoes vs. after!
Follow-up: Want to try one more? What single transformation do you get if you reflect in y = −x and then rotate 90° counterclockwise-and how does it change if you reverse the order?
Reflect in y = x swaps coordinates (x, y) → (y, x) and a 90° CCW rotation is (x, y) → (−y, x), so doing them in that order gives (x, y) → (−x, y), hence P→(−2, −3), Q→(−4, 1), R→(0, 0); reversing the order gives (x, y) → (x, −y), so the images differ and the order matters. A quick check I rely on is the 2×2 matrices S=[[0,1],[1,0]] (reflect y=x) and R=[[0,−1],[1,0]] (90° CCW), with R·S=[[-1,0],[0,1]] and S·R=[[1,0],[0,−1]]-I finally stopped second‑guessing after keeping these on a sticky note and just multiplying to confirm the order.
Short rules you can trust
– Reflect in the line y = x: (x, y) → (y, x).
– Rotate 90° counterclockwise about the origin: (x, y) → (−y, x).
– Reflections and rotations do not, in general, commute. Order matters.
Your example: reflect in y = x, then rotate 90° CCW
1) Start with a general point (x, y).
– After reflecting in y = x: (y, x).
– Then rotate 90° CCW: (−x, y).
So the combined effect is (x, y) → (−x, y). That’s simply a reflection in the y-axis.
Apply this to the triangle:
– P(2, −3) → (−2, −3)
– Q(4, 1) → (−4, 1)
– R(0, 0) → (0, 0)
If you swap the order (rotate first, then reflect in y = x)
1) Start with (x, y).
– Rotate 90° CCW: (−y, x).
– Reflect in y = x: (x, −y).
So the combined effect is (x, y) → (x, −y), which is a reflection in the x-axis.
Apply once to see the difference:
– P(2, −3) → (2, 3) (not the same as (−2, −3))
Hence, the order matters here.
Fixing the two mistakes in your attempt
– Reflection in y = x is not (x + y, x − y); it is just swapping the coordinates: (x, y) → (y, x).
– A 90° rotation is never the identity unless the point is at the origin. The correct 90° CCW rule is (x, y) → (−y, x).
A quick, reliable 2×2 matrix method
– Write each transformation as a 2×2 matrix acting on column vectors.
– Reflect in y = x: S = [[0, 1], [1, 0]].
– Rotate 90° CCW: R = [[0, −1], [1, 0]].
– Composition “reflect then rotate” corresponds to the product R·S (note the order: the rightmost matrix acts first):
– R·S = [[0, −1], [1, 0]] · [[0, 1], [1, 0]] = [[−1, 0], [0, 1]].
– That matrix sends (x, y) to (−x, y): reflection in the y-axis.
– The reversed order S·R = [[1, 0], [0, −1]] is reflection in the x-axis.
This matrix approach is not overkill. It is the cleanest way to combine transformations and avoids re-deriving rules.
Simple worked example (single point)
– Start with A(1, 2).
– Reflect then rotate: (1, 2) → (2, 1) → (−1, 2).
– Rotate then reflect: (1, 2) → (−2, 1) → (1, −2).
Different results, as expected.
Handy cheatsheet
– Reflect in x-axis: (x, −y).
– Reflect in y-axis: (−x, y).
– Reflect in y = x: (y, x).
– Rotate 90° CCW: (−y, x); 90° CW: (y, −x); 180°: (−x, −y).
For a clear overview of composing transformations with matrices, see Khan Academy: https://www.khanacademy.org/math/linear-algebra/matrix-transformations/compositions-of-transformations/a/compositions-of-linear-transformations