Y Plus Calculator

A Y+ coordinate describes the vertical component of a point or vector after its coordinate axes have been rotated. This calculator applies the standard two-dimensional rotation matrix to an original X and Y pair. It reports the transformed X coordinate, the transformed positive-axis Y component, and the vector magnitude. These results are useful when resolving forces, changing drawing orientations, analyzing motion, or translating measurements between local and global coordinate systems. For a counterclockwise angle θ, the transformed coordinates are x′ = x cos θ − y sin θ and y′ = x sin θ + y cos θ. Angles entered in degrees are converted to radians before the trigonometric functions are evaluated. A positive angle rotates counterclockwise; a negative angle rotates clockwise. Because a pure rotation does not stretch a vector, its magnitude √(x² + y²) remains unchanged. That invariant provides a quick way to check the calculation. Coordinate transformations appear throughout physics and engineering. A force measured along a tilted component can be converted into horizontal and vertical components. Surveying points can be aligned with a new baseline. Computer graphics uses the same matrix to rotate vertices, while robotics uses related matrices to express positions in different frames. The transformed Y+ value is simply the component directed along the positive Y axis of the selected frame; it may still be negative when the rotated point lies below that axis. Use consistent units for X and Y. If X is measured in meters, Y must also be measured in meters, and both transformed coordinates and the magnitude will be in meters. The angle is dimensionless but must be supplied in degrees for this calculator. Small floating-point values near zero can appear after rotations such as 90 or 180 degrees; the displayed result rounds those numerical artifacts to six decimal places. This tool performs a planar rotation about the origin. To rotate around another point, first subtract that point’s coordinates, rotate the resulting vector, and then add the center coordinates back.