Exercise 1: If a, b, and c are not all zero then ax + by + cz + d = 0 is a plane

In this video I start off with the linear equation of a plane and go backwards to show that, if at least one of a, b, and c are not zero, then the equation does indeed represent a plane. I show this by first assuming a is not equal to zero and writing the d term as (a d/a). Then I rewrite the equation as a dot product and simplify it further to show that it is just the vector equation of a plane. This is a good exercise and checkup on our earlier derivation of the linear equation of a plane.

The timestamps of key parts of the video are listed below:

- Exercise 1 + Hint: 0:00
- Solution: 0:46
- Rewriting Equation as a Dot Product: 1:55
- Obtaining a Vector Equation of a Plane: 3:54

This video was taken from my earlier video listed below:

- Equations of Lines and Planes: https://youtu.be/qWQz6qPhXR8
- Video notes: https://peakd.com/hive-128780/@mes/equations-of-lines-and-planes
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FO3u0IupqllNNkZTsffpIV

Related videos:

Vectors and the Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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