Find the intersection of $8x + 8y +z = 35$ and
$x = \left(\begin{array}{cc} 6\\ -2\\ 3\\ \end{array}\right) +$ $ \lambda_1 \left(\begin{array}{cc} -2\\ 1\\ 3\\ \end{array}\right) +$ $ \lambda_2 \left(\begin{array}{cc} 1\\ 1\\ -1\\ \end{array}\right) $
So, I have been trying this two different ways. One is to convert the vector form to Cartesian (the method I have shown below) and the other was to convert the provided Cartesian equation into a vector equation and try to find the equation of the line that way, but I was having some trouble with both methods.
Converting to Cartesian method:
normal = $ \left(\begin{array}{cc} -4\\ 1\\ -3\\ \end{array}\right) $
Cartesian of x $=-4x + y -3z = 35$
Solving simultaneously with $8x + 8y + z = 35$, I get the point $(7, 0, -21)$ to be on both planes, i.e., on the line of intersection.
Then taking the cross of both normals, I get a parallel vector for the line of intersection to be $(25, -20, -40)$.
So, I would have the vector equation of the line to be:
$ \left(\begin{array}{cc} 7\\ 0\\ -21\\ \end{array}\right) +$ $\lambda \left(\begin{array}{cc} 25\\ -20\\ -40\\ \end{array}\right) $
But my provided answer is:
$ \left(\begin{array}{cc} 6\\ -2\\ 3\\ \end{array}\right)+ $ $ \lambda \left(\begin{array}{cc} -5\\ 4\\ 8\\ \end{array}\right) $
I can see that the directional vector is the same, but why doesn't the provided answer's point satisfy the Cartesian equation I found?
Also, how would I do this if I converted the original Cartesian equation into a vector equation? Would I just equate the two vector equations and solve using an augmented matrix? I tried it a few times but couldn't get a reasonable answer, perhaps I am just making simple errors, or is this not the correct method for vector form?