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: https://peakd.com/hive-128780/@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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