9.3 Position Relations of Lines and Planes

For planes

\begin{equation*} \Pi_1: A_1x+B_1y+C_1z+D_1=0,\quad \Pi_2: A_2x+B_2y+C_2z+D_2=0, \end{equation*}

let \(\mathbf{n}_1=(A_1,B_1,C_1)\text{,}\) \(\mathbf{n}_2=(A_2,B_2,C_2)\text{.}\)

\(\mathbf{n}_1\parallel\mathbf{n}_2\) and all coefficients proportional \(\Rightarrow\) identical planes.
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\(\mathbf{n}_1\parallel\mathbf{n}_2\) but constants not in the same ratio \(\Rightarrow\) parallel distinct planes.
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\(\mathbf{n}_1\) not parallel to \(\mathbf{n}_2\) \(\Rightarrow\) planes intersect in a line.
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For lines

\begin{equation*} \ell_1:\mathbf{r}=\mathbf{r}_1+t\mathbf{v}_1,\quad \ell_2:\mathbf{r}=\mathbf{r}_2+s\mathbf{v}_2, \end{equation*}

we classify as follows:

Parallel if \(\mathbf{v}_1\parallel\mathbf{v}_2\text{.}\)
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Intersecting if there exist \(t,s\) such that \(\mathbf{r}_1+t\mathbf{v}_1=\mathbf{r}_2+s\mathbf{v}_2\text{.}\)
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Skew if they are not parallel and do not intersect.
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Determine the relation between

\begin{equation*} \ell_1:(x,y,z)=(1,0,2)+t(2,-1,1), \end{equation*}

\begin{equation*} \ell_2:(x,y,z)=(3,-1,1)+s(1,1,2). \end{equation*}

Solution.

Set coordinates equal:

\begin{equation*} \begin{cases} 1+2t=3+s,\\ -t=-1+s,\\ 2+t=1+2s. \end{cases} \end{equation*}

The augmented matrix is

\begin{equation*} \left[ \begin{array}{cc|c} 2 \amp -1 \amp 2\\ -1 \amp -1 \amp -1\\ 1 \amp -2 \amp -1 \end{array} \right]. \end{equation*}

Row reducing to reduced row echelon form gives

\begin{equation*} \left[ \begin{array}{cc|c} 1 \amp 0 \amp 1\\ 0 \amp 1 \amp 0\\ 0 \amp 0 \amp 0 \end{array} \right]. \end{equation*}

So \(t=1\text{,}\) \(s=0\text{.}\) Therefore the lines intersect at

\begin{equation*} (3,-1,3). \end{equation*}

Let a line have direction vector \(\mathbf{v}\text{,}\) and let a plane have normal vector \(\mathbf{n}\text{.}\)

If \(\mathbf{v}\cdot\mathbf{n}\neq 0\text{,}\) then the line intersects the plane at exactly one point.
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If \(\mathbf{v}\cdot\mathbf{n}=0\text{,}\) then the line is parallel to the plane or lies in the plane; substitute one point on the line into the plane equation to distinguish.
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