I need to find the coordinates of the intersection of the following plane and line through (0,0,0):
Translation: "en" means "and"
I do this by writing out the equations, claiming them to be equal 
and thus creating a matrix to solve for $\lambda, \mu, \sigma $:
So $\lambda=3, \mu=0, \sigma=4$, but this isn't the answer as the equations are NOT equal for those values. I really do not see where I go wrong though…
I expected not to find any solution, as the line and the plane are parallel. Is the fact that the solution I find is not a real answer to the problem an indication of this? I do not think so. I think there is a mistake in rewriting the equations to the matrix, but I really do not see what goes wrong.
This all should be very easy 🙁 But even after 1 hour I do not see my mistake. I would really appreciate some input on my line of thought.
Best Answer
The system of equations you're trying to express as a matrix operation is:
$\sigma\left( \begin{array}{c} 1\\ -2\\ -4\\ \end{array} \right)=\left( \begin{array}{c} 3\\ 0\\ -2\\ \end{array} \right)+\lambda\left( \begin{array}{c} 1\\ 1\\ 1\\ \end{array} \right)+\mu\left( \begin{array}{c} 2\\ -1\\ -3\\ \end{array} \right)$
Hence, $ \begin{pmatrix} 1 & 2 & -1 \\ 1 & -1 & 2 \\ 1 & -3 & 4 \end{pmatrix} \begin{pmatrix} \lambda \\ \mu \\ \sigma \end{pmatrix} =\begin{pmatrix} -3 \\ 0 \\ 2 \end{pmatrix}$
Row reducing (what you were referring to as sweeping, I believe) will give you solutions for $\lambda, \mu,$ and $\sigma$. In order to find $\textbf{x}$, we use the fact that
$\sigma\begin{pmatrix} 1 \\ -2 \\ -4 \end{pmatrix}=\textbf{x}$