Determine the relative positions of two lines [latex]S^{1}_{1} =\overrightarrow{x_{0} }+S_{1} and S^{1}_{2} =\overrightarrow{y_{0} }+S_{2} [/latex] in R³ (see Fig. 3.13).

Remind that both [latex]S_{1} [/latex] and [latex]S_{2} [/latex] are one-dimensional vector subspaces of R³ and [latex]\overrightarrow{y_{0} }-\overrightarrow{x_{0} }[/latex] is a vector in R³. In case [latex]S_{1}=S_{2} [/latex] : then [latex]\overrightarrow{x_{0} }=\overrightarrow{y_{0} }[/latex] will induce that [latex]S^{1}_{1}[/latex] is coincident with [latex]S^{1}_{2},[/latex] while [latex]\overrightarrow{x_{0} }\neq \overrightarrow{y_{0} }[/latex] will result in the parallelism of [latex]S^{1}_{1}[/latex] with [latex]S^{1}_{2}.[/latex] In case [latex]S_{1} \cap S_{2} = \left\{\overrightarrow{0}\right\}:[/latex] then the condition [latex]\overrightarrow{x_{0} }-\overrightarrow{y_{0} } \in S_{1}\oplus S_{2}[/latex] implies that [latex]S^{1}_{1}[/latex] intersects [latex]S^{1}_{2}[/latex] at a point, while [latex]\overrightarrow{x_{0} }-\overrightarrow{y_{0} } \notin S_{1}\oplus S_{2}[/latex] implies that [latex]S^{1}_{1}[/latex] is skew to[latex]S^{1}_{2}.[/latex]

Question 3.36: Determine the relative positions of two lines S11 = x0^→ + S...

Geometric Linear Algebra

Determine the relative positions of two lines $S^{1}_{1} =\overrightarrow{x_{0} }+S_{1} and S^{1}_{2} =\overrightarrow{y_{0} }+S_{2}$ in R³ (see Fig. 3.13).

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Remind that both $S_{1}$ and $S_{2}$ are one-dimensional vector subspaces of R³ and $\overrightarrow{y_{0} }-\overrightarrow{x_{0} }$ is a vector in R³.

In case $S_{1}=S_{2}$ : then $\overrightarrow{x_{0} }=\overrightarrow{y_{0} }$ will induce that $S^{1}_{1}$ is coincident with $S^{1}_{2},$ while $\overrightarrow{x_{0} }\neq \overrightarrow{y_{0} }$ will result in the parallelism of $S^{1}_{1}$ with $S^{1}_{2}.$

In case $S_{1} \cap S_{2} = \left\{\overrightarrow{0}\right\}:$ then the condition $\overrightarrow{x_{0} }-\overrightarrow{y_{0} } \in S_{1}\oplus S_{2}$ implies that $S^{1}_{1}$ intersects $S^{1}_{2}$ at a point, while $\overrightarrow{x_{0} }-\overrightarrow{y_{0} } \notin S_{1}\oplus S_{2}$ implies that $S^{1}_{1}$ is skew to $S^{1}_{2}.$