The projection formula is $\operatorname{proj}_u(v)=u\,\langle u,v\rangle/\|u\|^2$ for $u\ne0$. The projection matrix is $uu^*/(u^*u)$. For real $u=(1,1)$ and $v=(2,0)$ it gives $(1,1)$, leaving residual $(1,-1)$ orthogonal to $u$.
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