Htykuut

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.

The system of equations you're trying to express as a matrix operation is:

$sigmaleft(
begin{array}{c}
1\
-2\
-4\
end{array}
right)=left(
begin{array}{c}
3\
0\
-2\
end{array}
right)+lambdaleft(
begin{array}{c}
1\
1\
1\
end{array}
right)+muleft(
begin{array}{c}
2\
-1\
-3\
end{array}
right)$

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

$sigmabegin{pmatrix}
1 \
-2 \
-4
end{pmatrix}=textbf{x}$