Slope of a Line Segment between Two Points in Three Dimensions

The geometric intuition for a slope is or our rise for every unit that we move.

Example: Find the slope of the segment from (1, 1, 0) to (4, 5, 6).

Solution:

  1. Find the run: The run is between (1, 1, 0) and (4, 5, 0) in the xy



    Using Pythagoras, we can conclude that the run is 5.

  2. Find the rise:



    The rise is the distance from (4, 5, 0) to (4, 5, 6) is 6.

  3. To visualize the slope we are obtaining, place the right triangle with vertices (1, 1, 0), (4, 5, 0), and (4, 5, 6) in 3-space.



  4. The slope is


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