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6 Responses
Love this pattern: (3,1) → (−1,3) matches (x,y) → (−y,x), which is a 90° counterclockwise rotation-your matrix [[0,-1],[1,0]] is exactly the right one. Quick check: v·w=0 gives 90°, and det[v w]=10>0 says CCW; want a neat mnemonic to remember these tests?
Ooh, classic rotation-spotting! Your matrix is totally relevant-[[0, -1], [1, 0]] is the standard 90° counterclockwise rotation, and applying it to (3, 1) gives (-1, 3), so this is a 90° CCW turn. A neat way to recognize these quickly is to remember the patterns: (x, y) -> (-y, x) is 90° CCW, (x, y) -> (y, -x) is 90° clockwise (aka 270° CCW), and (x, y) -> (-x, -y) is 180°. If you want a more “calculation-proof” check, use dot and cross: v·v’ = 3·(-1) + 1·3 = 0, so the angle is 90°, and x y’ − y x’ = 3·3 − 1·(−1) = 10 > 0, which means the turn is counterclockwise. Want a quick mnemonic or a visual trick to remember those sign-and-swap patterns for 90°, 180°, and 270°?
Nice! Your matrix is exactly the right tool here. The rule (x, y) -> (-y, x) is the 90° counterclockwise rotation about the origin, and your matrix [[0, -1], [1, 0]] produces exactly that. Applying it to (3, 1) gives (-1, 3), so-unless I’m missing a sneaky sign-this is a 90° counterclockwise rotation. For contrast, a 90° clockwise (which is 270° counterclockwise) would be (x, y) -> (y, -x), sending (3, 1) to (1, -3), which doesn’t match.
If you want a clean, matrix-free check: the image should have the same length and be perpendicular. Indeed, (3, 1) and (-1, 3) both have length sqrt(10), and their dot product is 3·(-1) + 1·3 = 0, so it’s a right-angle turn. To decide the direction, compute the determinant x1*y2 − y1*x2 = 3*3 − 1*(-1) = 10 > 0; positive means a counterclockwise turn. So the rotation is +90° (counterclockwise), not −90° (clockwise).
Great question! For rotations about the origin, there are handy “pattern” rules you can memorize:
– 90° counterclockwise: (x, y) → (−y, x)
– 90° clockwise (same as 270° counterclockwise): (x, y) → (y, −x)
– 180°: (x, y) → (−x, −y)
Your point (3, 1) became (−1, 3). That matches (−y, x), so it’s a 90° counterclockwise rotation. It’s not 270° counterclockwise (or 90° clockwise), because that would give (1, −3). If you like matrices, you were right on target: the matrix [[0, −1], [1, 0]] is exactly the 90° counterclockwise rotation, and applying it to [3; 1] gives [−1; 3]. The 270° counterclockwise matrix would be [[0, 1], [−1, 0]].
Quick worked example: take (2, 5).
– 90° counterclockwise: (2, 5) → (−5, 2). Using the matrix [[0, −1], [1, 0]]: [0·2 − 1·5, 1·2 + 0·5] = [−5, 2].
– 90° clockwise: (2, 5) → (5, −2).
– 180°: (2, 5) → (−2, −5).
Spotting which pattern you have makes the angle and direction clear.
Spin the point like a pizza slice: 90° CCW sends (x, y) → (−y, x) (your matrix [[0, −1], [1, 0]] is exactly this), while 90° CW sends (x, y) → (y, −x); since (3, 1) → (−1, 3), it’s a 90° counterclockwise rotation (aka 270° clockwise). Quick refresher: https://www.khanacademy.org/math/geometry-home/geometry-transformations/rotations/v/rotating-shapes-90-degrees-1 – hope this helps!
Yes-your matrix [0 -1; 1 0] is the 90° counterclockwise rotation, since it sends (x, y) → (−y, x), so (3, 1) → (−1, 3) (aka 270° clockwise). Want a quick trick to remember which sign flips for CW vs CCW?