Problems Plus 3: Line and Surface of Two Intersecting Planes

In this video I determine the symmetric equations of a line at the intersection of two planes. I then find out that when the equations of the plane vary, the line sweeps out a circular hyperboloid of one sheet. Finally, I calculate the volume of this surface between two planes.

Timestamps:

• Problem 3: Intersecting planes: 0:00
• Solution to (a): Symmetric equations for L: 1:07
  • Finding a point on both planes: 1:52
  • Obtaining normal vectors of the planes: 5:20
  • Cross product of normal vectors is parallel to L: 7:14
  • Obtain symmetric equations of L: 13:02
  • Symmetric equations of L formula: 15:30
  • Case where c = 0: 16:53
  • Case where c = -1: 18:18
  • Case where c = 1: 20:54
  • Graphing with GeoGebra: 22:52
• Solution to (b): Curve of intersection: 28:16
  • Solving y and x in terms of c and t: 30:34
  • Eliminating c by finding squares of x and y: 36:15
  • Obtain a trigonometric identity: 39:26
  • Using trigonometric identity to eliminate c: 40:20
  • Obtain a circle: 44:26
  • Graphing in GeoGebra shows a hyperboloid of one sheet: 45:11
• Solution to (c): Volume of the hyperboloid: 49:26
  • Volume is integral of area = 4/3 pi: 52:21

Video notes and playlists:

• HIVE notes: @mes/jlgigkdl
• Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0G_7Jd6ZORHFQMk8jbwmkos
• Vectors and the Geometry of Space playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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