Orthogonal Projection onto a Subspace
How to find the orthogonal projection of a vector y onto a subspace W?
A popular method is to use an orthogonal basis in W.
Then, the output is the sum of the orthogonal projections of y onto the coordinate axes.
This applet also illustrates the formula for the Fourier coefficients, see the projections of y on the coordinate axes.
Remark: This is a modified version of the applet https://www.geogebra.org/m/zxxfqa4t by Trefor Bazett