Lines
Definition of lines and determining positional relationships.
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Study Lines
Representing lines
Idea: A line is uniquely determined if we know a point and the direction of the line.
Definition 1. Every line in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) can be described in the following form: \[g=\left\{\mathbf{p}+\lambda \mathbf{a}:\ \lambda\in\mathbb{R}\right\} = \mathbf{p} + \text{Span}(\mathbf{a}),\] where the points of \(g\) are identified with their position vectors. This description is called parameter form. The vector \(\mathbf{p}\) represents an arbitrary point of \(g\), and \(\mathbf{a}\) represents the direction of the line.
Example 2. The line \(g\), which goes through the points \[A:=(0,1,2) \quad \text{and} \quad B:=(-2,7,10),\] has the parameter form \[g= \left\{\begin{pmatrix}0\\1\\2\end{pmatrix} + \lambda \begin{pmatrix}-2-0\\7-1\\10-2\end{pmatrix}: ~ \lambda \in \mathbb{R}\right\}.\]
Determining positional relationships for lines
Let two lines \(g_1\) and \(g_2\) be given by a parameter form: \[\begin{aligned} g_1 & = \{ \mathbf{p}_1 + \lambda_1 \mathbf{r}_1:~ \lambda_1\in\mathbb{R} \} \, ,\\ g_2 & = \{ \mathbf{p}_2 + \lambda_2 \mathbf{r}_2:~ \lambda_2\in\mathbb{R} \} \, . \end{aligned}\] In order to determine the positional relationship, both representations can be equalized, resulting in a system of linear equations (with variables \(\lambda_1,\lambda_2\)): \[\mathbf{p}_1 + \lambda_1 \mathbf{r}_1 = \mathbf{p}_2 + \lambda_2 \mathbf{r}_2\] There are four cases:
| exactly one solution | intersection point (put in \(\lambda_1\), \(\lambda_2\) into the parameter forms) |
|---|---|
| infinitely many solutions | identical lines |
| no solution \((\lambda_1,\lambda_2)\) & \(\mathbf{r}_1\) and \(\mathbf{r}_2\) are multiples of each other | lines are parallel but not identical |
| no solution \((\lambda_1,\lambda_2)\) & \(\mathbf{r}_1\) and \(\mathbf{r}_2\) are no multiples | skew lines |
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