(a) To find the equation of \(\Pi_1\), we need a normal vector to the plane. Since \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\), the normal vector can be found using the cross product of the direction vectors of \(l_1\) and \(l_2\):
\(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 1 & 1 \\ 1 & -4 & 2 \end{vmatrix} = \begin{pmatrix} 6 \\ -3 \\ -9 \end{pmatrix}\)
Substitute the point \((1, 3, -2)\) from \(l_1\) into the plane equation \(6x - 3y - 9z = d\) to find \(d\):
\(6(1) - 3(3) - 9(-2) = d\)
\(6 - 9 + 18 = d\)
\(d = 15\)
Thus, the equation of \(\Pi_1\) is \(2x - y - 3z = 5\).
(b) The normal to \(\Pi_2\) is the direction vector of \(l_2\), \(\begin{pmatrix} 1 \\ -4 \\ 2 \end{pmatrix}\). The angle \(\theta\) between \(\Pi_1\) and \(\Pi_2\) is given by:
\(\cos \theta = \frac{\mathbf{n_1} \cdot \mathbf{n_2}}{\|\mathbf{n_1}\| \|\mathbf{n_2}\|}\)
\(\cos \theta = \frac{\begin{pmatrix} 6 \\ -3 \\ -9 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ -4 \\ 2 \end{pmatrix}}{\sqrt{6^2 + (-3)^2 + (-9)^2} \sqrt{1^2 + (-4)^2 + 2^2}}\)
\(\cos \theta = \frac{-7}{\sqrt{14 \times 77}}\)
\(\theta = 77.7^\circ\)
(c) To find \(\mathbf{PQ}\), use the points \(\mathbf{P}\) and \(\mathbf{Q}\) on \(l_1\) and \(l_2\) respectively:
\(\mathbf{OP} = \begin{pmatrix} 1 + 2\lambda \\ 3 + \lambda \\ -2 + \lambda \end{pmatrix}\)
\(\mathbf{OQ} = \begin{pmatrix} 1 + \mu \\ -2 - 4\mu \\ 9 + 2\mu \end{pmatrix}\)
\(\mathbf{PQ} = \mathbf{OQ} - \mathbf{OP} = \begin{pmatrix} \mu - 2\lambda \\ -5 - 4\mu - \lambda \\ 11 + 2\mu - \lambda \end{pmatrix}\)
Set \(\mathbf{PQ}\) perpendicular to both direction vectors:
\(\begin{pmatrix} \mu - 2\lambda \\ -5 - 4\mu - \lambda \\ 11 + 2\mu - \lambda \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} = 0\)
\(\begin{pmatrix} \mu - 2\lambda \\ -5 - 4\mu - \lambda \\ 11 + 2\mu - \lambda \end{pmatrix} \cdot \begin{pmatrix} 1 \\ -4 \\ 2 \end{pmatrix} = 0\)
Solve these equations to find \(\lambda\) and \(\mu\), then substitute back to find \(\mathbf{PQ}\):
\(\mathbf{r} = \begin{pmatrix} -1 \\ 6 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 6 \\ -3 \\ -9 \end{pmatrix}\)
