Write the projection of the line ell : 4x-y+3z-6=0…
Write the projection of the line $\ell$: $4x-y+3z-6=0$, $x+5y-z+10=0$ onto the plane $\pi$: $2x-y+5z-5=0$.
Explanation of Method
The orthogonal projection of a line onto a plane, via the projecting plane.
Step 1: Projecting plane
This is the member of the pencil through $\ell$ that is perpendicular to $\pi$: $(4+\lambda)x+(5\lambda-1)y+(3-\lambda)z+(10\lambda-6)=0$.
\[ 2(4+\lambda)-(5\lambda-1)+5(3-\lambda)=24-8\lambda=0\ \Longrightarrow\ \lambda=3:\quad 7x+14y+24=0 . \]
Step 2: Projection
\[\ell':\ \begin{cases}7x+14y+24=0\\ 2x-y+5z-5=0\end{cases}\]
Direction of $\ell'$: $\{7,14,0\}\times\{2,-1,5\}\parallel\{2,-1,-1\}$, which lies in $\pi$ ($4+1-5=0$). The point $\left(\tfrac{20}{21},-\tfrac{46}{21},0\right)$ of $\ell$ lies in the projecting plane: $\tfrac{140-644+504}{21}=0$.
Result
\[\boxed{\,\ell':\ \begin{cases}7x+14y+24=0\\ 2x-y+5z-5=0\end{cases}\,}\]
Remark
The source's attempted point with $z=0$ was abandoned halfway (“$y=-\tfrac{46}{21}\ldots$”). It works: $x=-10-5y=\tfrac{20}{21}$.