11c. Lines and planes

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Consider the points $A(3,-1,6), B(1,-1,-4)$ and $C(5,3,0)$.

Consider the points $A(3,-1,6), C(5,3,0), D(5,-2,5)$ and the lines
$l_1: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ l_2: \mathbf{r} =\begin{pmatrix} 1 \\ -1 \\ -4 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix}, \mu \in \mathbb{R}$.

The lines $l$ and $l_3$ and the planes $\pi_1$ and $\pi_2$ have equations
$l_1: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ l_3: \mathbf{r} = \begin{pmatrix} 5 \\ 3 \\ 0 \end{pmatrix} + \nu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}, \nu \in \mathbb{R} \\ \pi_1: x+y+z = 8 \\ \pi_2: 2x-3y+5z=-2$.

Consider the lines $l$ and $m$ and the planes $\pi_1$ and $\pi_2$ with equations:
$l: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ m: \mathbf{r} = \begin{pmatrix} 5 \\ 4 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}, \mu \in \mathbb{R} \\ \pi_1: x+y+z = 8 \\ \pi_2: 2x-3y+5z=-2$.

The point $B$ has coordinates $(1,-1,4)$. The equations of the line $l$ and the plane $\Pi$ are:
$l: \mathbf{r}=(3\mathbf{i}-\mathbf{j}+6\mathbf{k}) + \lambda (\mathbf{i}+2\mathbf{j}-3\mathbf{k}), \lambda \in \mathbb{R} \\ \Pi: 2x-3y+5z = -2$.

The point $B$ has coordinates $(1,-1,4)$. The equations of the line $l$ and the plane $\pi$ are:
$l: \mathbf{r}=(3\mathbf{i}-\mathbf{j}+6\mathbf{k}) + \lambda (\mathbf{i}+2\mathbf{j}-3\mathbf{k}), \lambda \in \mathbb{R} \\ \pi: x+y+z = 8$.

Consider the lines $l, m_1, m_2$ and $m_3$ and the planes $\Pi_1$ and $\Pi_2$ with equations: $l: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ m_1: \mathbf{r} =\begin{pmatrix} 7 \\ 6 \\ -5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 3 \\ -5 \end{pmatrix}, \mu \in \mathbb{R} \\ m_2: \mathbf{r} =\begin{pmatrix} 4 \\ 0 \\ 7 \end{pmatrix} + \nu \begin{pmatrix} 2 \\ 3 \\ -5 \end{pmatrix}, \nu \in \mathbb{R} \\ m_3: \mathbf{r} =\begin{pmatrix} 4 \\ 0 \\ 7 \end{pmatrix} + \omega \begin{pmatrix} -2 \\ 4 \\ 6 \end{pmatrix}, \omega \in \mathbb{R} \\ \Pi_1: ax+by = 4 \\ \Pi_2: cx+y-z=d$.