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