The formula for rotation 90 degrees clockwise about the origin in a standard Cartesian coordinate plane is (x, y) → (y, −x). This rule swaps the coordinates and changes the sign of the original x-coordinate, producing a quarter-turn without changing the point’s distance from the center of rotation. It is a fundamental transformation used in geometry, computer graphics, engineering drawings, and coordinate-based problem solving.
Introduction
A rotation turns a point or shape around a fixed point called the center of rotation. A 90-degree rotation creates a quarter-turn, and the word clockwise indicates the same direction in which the hands of a clock move.
In the usual Cartesian plane, positive x values extend to the right and positive y values extend upward. Under this convention, a point initially to the right of the origin moves downward
to (0, −x), assuming (x>0). That movement illustrates why the new (y)-coordinate must be negative for points originally on the positive x-axis Small thing, real impact. And it works..
How the transformation works
Consider the point ((3,2)). Applying the clockwise rotation gives:
[ (3,2)\rightarrow(2,-3) ]
The coordinates were first swapped, and then the original (x)-coordinate, (3), was given a negative sign. The result lies in the fourth quadrant.
This transformation preserves both the distance from the origin and the shape’s size. For any point ((x,y)),
[ x^2+y^2=y^2+(-x)^2, ]
so its distance from the origin remains unchanged. The dot product of the original and rotated vectors is also zero:
[ (x,y)\cdot(y,-x)=xy-xy=0, ]
which confirms that the two vectors are perpendicular and therefore separated by (90^\circ) No workaround needed..
Step-by-step procedure
To rotate a point (90^\circ) clockwise about the origin:
- Identify the original coordinates ((x,y)).
- Swap (x) and (y).
- Change the sign of the value that was originally (x).
- Plot the resulting point ((y,-x)).
For example:
[ (-4,7)\rightarrow(7,4) ]
Here, the coordinates are swapped to ((7,-4)), and then (-4) becomes (4).
Examples
| Original point | Clockwise rotation |
|---|---|
| ((5,1)) | ((1,-5)) |
| ((-2,6)) | ((6,2)) |
| ((0,3)) | ((3,0)) |
| ((-4,-1)) | ((-1,4)) |
| ((7,0)) | ((0,-7)) |
A point on an axis may appear to lose one of its nonzero coordinates after rotation, but the distance from the origin remains the same.
Rotating a figure
To rotate a polygon, apply the rule to every vertex, plot the new vertices, and connect them in the same order. To give you an idea, suppose a triangle has vertices
[ A(1,2),\quad B(4,2),\quad C(2,
