Determine the matrix $F$ in ${\rm I\!R}^3$ that is defined by first projecting $u$ on the plane through origo with the normal $(1,2,1)\times(1,0,1)$ and then the picture of $P(u)$ gets projected on $P(u)\times v$ where $v=(1,1,1)$. (Assume ON-base)
So here is how I thought to solve it:
First, let $N=(1,2,1)\times(1,0,1)=(2,0,-2)$ we can then use the projection formula to see what happens with our base-vector $(e_1, e_2, e_3)$.
So $$e_{1//N}=\frac{e_1\cdot N}{|N|^2}\cdot N=\frac{2}{16}(2,0,-2)$$ $$e_{2//N}=\frac{e_2\cdot N}{|N|^2}\cdot N=0\cdot(2,0,-2)$$ $$e_{2//N}=\frac{e_3\cdot N}{|N|^2}\cdot N=\frac{-2}{16}(2,0,-2)$$ This is the part where I'm note completly sure, my thought is that this gives us the matrix $$P(u)=(e_{1//N},e_{2//N},e_{3//N})$$ From here I'm not quite sure how to proceed. My thought is to multiply $u\cdot P(u)$ and we will get, let's call it $w$ and then $w\times v$. But since we are looking for the matrix $F$ this can't be the right approach.
Looking for some guidance in how to approach these kinds of problems,
Many thanks,