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  5. Intersection of Lines and Planes

Intersection of Lines and Planes

A jet is approaching a busy airport. Although the pilot may not physically see the airport yet, the jet’s path is a straight line aimed at the flat surface of the runway. Electronic navigation aids help the pilot and air traffic controller guide the jet to a safe landing. Being able to predict whether the flight path will land the jet on the runway at the correct time is crucial.

Investigate: Intersections of Lines and Planes in Three-Space

  • Use a straw and a piece of cardboard to represent a line and a plane in three-space.
  • Consider the following orientations:
    • Single intersection: The line meets the plane at one point.
    • Infinite intersections: The line lies on the plane.
    • No intersections: The line is parallel to the plane.

Reflect: How are the direction vectors of the lines and the normal vectors of the planes related in each scenario?

Key Concepts

  • A line and a plane in three-space can intersect in three ways:
    • The line intersects the plane at a point (one solution).
    • The line lies on the plane (infinitely many solutions).
    • The line is parallel to the plane and does not intersect (no solution).
  • The intersection of a line and a plane can be determined by substituting the parametric equations of the line into the plane’s scalar equation.

Example: Find the Intersection of a Line and a Plane

Determine if the line and plane intersect. If so, find the solution.

Given:

  • Plane: \(9x + 13y + 2z = 29\)
  • Line: \([x, y, z] = [5, -5, 2] + t[2, 5, 3]\)

Solution:

  1. Substitute the parametric equations of the line into the plane equation:

    2. \(9(5 + 2t) + 13(-5 + 5t) + 2(2 + 3t) = 29\)

    \(45 + 18t - 65 + 65t + 4 + 6t = 29\)

    \(53t = 53\)

    \(t = 1\)

  2. Find the intersection point by substituting \(t = 1\) into the parametric equations:

    3. \(x = 5 + 2(1) = 7\)

    \(y = -5 + 5(1) = 0\)

    \(z = 2 + 3(1) = 5\)

    Conclusion: The line and plane intersect at the point (7, 0, 5).

Summary

  • If substituting the parametric equations into the plane results in a valid value of \(t\), the line and plane intersect at one point.
  • If the result is always true (e.g., \(0 = 0\)), the line lies on the plane.
  • If the result is false (i.e. \(0 = 5\)), the line and plane are parallel.


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