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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

Expert verified

The two distinct parallel lines determine a unique plane.

Step by step solution

01

Given information

The two distinct parallel lines.

02

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.

03

Calculation

The following steps are followed to find the equation of the plane from a point Pnot on lineLand the line L

  • Choose any point Q on the line L
  • Construct the vector PQ
  • Find the normal vectorNby N=PQ×dwhere d is the direction vector of the line L
  • Use Pand Nto find the equation of the plane.

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