Question

Explain why two distinct parallel lines determine a unique plane. Explain how you would use the equations of the lines to find the equation of the plane.

Short answer

The two distinct parallel lines determine a unique plane.

Step-by-step solution

Given information

The two distinct parallel lines.

Calculation

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.

Calculation

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 $\overrightarrow{PQ}$
• Find the normal vector$\mathbf{N}$by $\mathbf{N}=\overrightarrow{PQ}\times \mathbf{d}$where $\mathbf{d}$ is the direction vector of the line $\mathfrak{L}$
• Use $P$and $\mathbf{N}$to find the equation of the plane.