Question

For 100 points, solve:



$\text{We are given two planes \Pi_1 and \Pi_2 in parametric vector form,}$

$\[\Pi_1 \ : \ \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}= \begin{pmatrix} 0 \\ -2 \\ 0 \end{pmatrix} + \lambda_1\begin{pmatrix} 1 \\ -1 \\ -2 \end{pmatrix} + \lambda_2\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix} \]$

$\[\Pi_2 \ : \ \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}= \begin{pmatrix} 0 \\ -2 \\ 4 \end{pmatrix} + \mu_1\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} + \mu_2\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}\]$

$\[\text{where x_1 , x_2 , x_3 , \lambda_1 , \lambda_2 , \mu_1 , and \mu_2\in\mathbb{R} }$

Find vectors n₁ and n₂ that are normals to π₁ and π₂ respectively and explain how you can tell without performing any extra calculations that π₁ and π₂ must intersect in a line.