91Ó°ÊÓ

The two distinct parallel lines.

The goal is to explain why two separate parallel lines determine a unique plane and how to derive the plane's equation using line equations.

Use the statement that a unique plane is determined by three non-collinear points to explain the outcome.

Choose two separate spots on one line and a third point on the other line. The three points that were picked are non-collinear.

A unique plane is determined by the three non-collinear points. The three points selected are noncollinear.

That's why the two distinct parallel lines determine a unique plane.

The following steps are followed to find the equation of the plane from a point $P$not on line$\mathfrak{L}$and the line $\mathfrak{L}$

• Choose any point $Q$ on the line $\mathfrak{L}$
• Construct the vector $\stackrel{→}{PQ}$
• Find the normal vector$\mathbf{N}$by $\mathbf{N}=\stackrel{→}{PQ}×\mathbf{d}$where $\mathbf{d}$ is the direction vector of the line $\mathfrak{L}$
• Use $P$and $\mathbf{N}$to find the equation of the plane.