Problems Plus 3: Line and Surface of Two Intersecting Planes

about 2 months ago

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