We have two non-parallell vectors $\vec{a}$ and $\vec{b}$ in three dimensional space, together with an arbitrary vector $\vec{c}$. I want to find the orthogonal projection $\vec{c'}$ of $\vec{c}$ on the plane that is spanned by $\vec{a}$ and $\vec{b}$, where $\vec{c'}$ is a linear combination of $\vec{a}$ and $\vec{b}$.
I have tried to approach this problem in a couple of ways.
My first thought was to project c onto the normal vector, $\vec{n}$, of the plane and then take $\vec{c}-proj_{n}\ \vec{c}$ which would give us a vector in the plane, i.e the projection $\vec{c'}$ of $\vec{c}$ onto the plane. However, with this approach, I cannot rewrite it as a linear combination of $\vec{a}$ and $\vec{b}$.
Another approach I have used is project $\vec{c}$ onto $\vec{a}$ and $\vec{b}$, respectively, and then add them together. With this approach I get it as a linear combination of $\vec{a}$ and $\vec{b}$, but this is not the right answer.
Any help and tips would be greatly appreciated!