Orthogonal Projections are a type of projection that maps one vector onto another:
![[Pasted image 20260211175349.png]]
In this image the result for the projection of $x$ onto $y$ is $x_p$
in order to derive this we can define $x_p = cy$ where $c$ is a mysterious scalar as $x_p$ is always in the same direction as $Y$ and we can define $x_o = x-x_p = x-cy$ and because $x_p$ is orthogonal we can define it as an Inner Product %%[[Inner Products]]%% like this: $⟨x − 𝑐y, y⟩ = 0$ then to solve for c like this: $$⟨x − 𝑐y, y⟩ = 0
$$$$⟨x, y⟩-𝑐⟨y, y⟩ = 0$$ $$⟨x, y⟩ = c⟨y, y⟩$$ $$\frac{⟨x, y⟩}{⟨y, y⟩} = 𝑐$$
so after all that we get $c=\frac{⟨x, y⟩}{⟨y, y⟩}$ from this we can derive $x_p$ by doing this: $$x_p = \frac{⟨x, y⟩}{⟨y, y⟩}y$$
because $x_p = cy$ and $c=\frac{⟨x, y⟩}{⟨y, y⟩}$.