I was given the following question:
Given $\{u_1,u_2\}$ is an orthogonal set, find the orthogonal projection of $y$ onto Span$\{u_1,u_2\}$.$$y=\left(\begin{matrix}-1\\3\\6\\\end{matrix}\right), u_1=\left(\begin{matrix}-5\\-1\\2\\\end{matrix}\right),u_2=\left(\begin{matrix}1\\-1\\2\\\end{matrix}\right)$$
I know how to find proj$_a\vec{b}$. It is $\frac{a\bullet b}{||a||^2}\vec{a}$. What is throwing me off is the fact that here I'm not looking for the projection of y onto a vector, I'm looking for the projection of y onto the span of two vectors. How do I deal with that?
It is the linear combination of those two )i.e., onto the plane spanned by those two orthogonal vectors):
$$Py=\frac{y\cdot u_1}{\left\|u_1\right\|^2}\,u_1+\frac{y\cdot u_2}{\left\|u_2\right\|^2}\,u_2$$