Line Parallel to Two Planes [modified from FM N2005/I/9 part]

The planes  $\Pi_1$ and $\Pi_2$  have vector equation
           $\mathbf{r} = \lambda_1(\mathbf{i + j - k}) + \mu_1 (2\mathbf{i - j + k})$
 and     $\mathbf{r} = \lambda_2(\mathbf{i + }2\mathbf{j + k}) + \mu_2 (3\mathbf{i + j - k})$
respectively.  The line  l  passes through the point with position vector $4\mathbf{i + }5\mathbf{j + }6\mathbf{k}$  and is parallel to both   $\Pi_1$ and $\Pi_2$.  Find a vector equation for  l.

Method 1
First, we find the normal vectors to the planes  $\Pi_1$ and $\Pi_2$ respectively.

Taking cross-products yields the direction parallel to both planes

Hence the required line parallel to both planes is

Method 2
Putting the two equations together

Transferring terms to the left side, one obtains the simultaneous system

Solving by Graphing Calculator, one obtains $\lambda_1 = \frac{5}{3}t$, $\mu_1 = \frac{2}{3}t$, $\lambda_2 = 0$, $\mu_1 = t$.  Upon substitution, we find The line of intersection is given by

Hence the required line parallel to both planes is