Example 6: Curve of Intersection of a Cylinder and Plane

In this video I go over an example on determining the vector function of the curve of intersection between a vertical cylinder and a slanted plane. I first find an equation of the intersection by plugging the equation of the plane into the equation of the cylinder. I then determine 4 points of intersection and manually graph it out, obtaining an ellipse. Projecting this ellipse onto the x-y plane obtains a unit circle of radius 1. By using the Pythagorean Identity, I obtain parametric equations of the circle as trigonometric functions for x and y. By plugging the y parametric equation into the plane equation, we obtain the z parametric equation. From here we can combine the parametric equations into a vector function. Note that earlier in the video, I also graphed the plane and cylinder in GeoGebra 3D graphing calculator: https://www.geogebra.org/calculator/v7ecchdb

#math #3D #geogebra #science #calculus

Timestamps:

- Example 6: Curve of Intersection: 0:00
- Graphing the cylinder and plane in GeoGebra: 0:20
- Equation at intersection: 1:23
- 2 Points of intersection when x = 0: 2:37
- 2 Points of intersection when x = +/- 1: 5:13
- Graphing out points of intersection: 7:23
- Better illustration of the intersection points from my Calculus book: 9:50
- Projection of the intersection curve onto x-y plane is a circle: 10:56
- Getting parametric equations of a circle via the Pythagorean Identity: 11:46
- Converting plane equation into trigonometric version: 13:26
- Parametric equations for the intersection curve: 14:27
- Corresponding vector equation of the intersection curve: 15:00
- Arrows on the curve indicate how the curve is traced as the parameter t increases: 16:03
- Parameter t is the angle going counterclockwise from the positive x axis: 17:16

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-skxajmlt
- Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GwdMY-E9_4LsmYmO-IxcoX
- Notes: https://peakd.com/hive-128780/@mes/omhbcxkw
- Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .

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