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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
Problems Plus 3: Line and Surface of Two Intersecting Planes