Orthogonal projection of $v=(x,y)$ on $x=y$

Question

I would like to find the orthogonal projection of a vector $v=(x,y)$ on $\beta: x=y$ in $\mathbb{R^2}$. I know that a basis for $\beta$ is $B=\{e_1+e_2=(1,1)\}$. The vector decomposition of a general vector $v$ is $v=\langle v, e_1\rangle \cdot e_1+\langle v, e_2\rangle \cdot e_2$. How can I proceed in finding the orthogonal projection?

Answer 1

First you normalise the vector $(1,1)$, thereby getting $\frac1{\sqrt2}(1,1)$. Then, for each $(x,y)\in\mathbb R^2$, the projection will be$$\left\langle(x,y),\frac1{\sqrt2}(1,1)\right\rangle\frac1{\sqrt2}(1,1)=\left(\frac{x+y}2,\frac{x+y}2\right).$$