Objectives
- To understand how rectangular coordinates are measured
Section 2.1 Rectangular 3D Coordinates
In this section, rectangular coordinates are introduced in terms of three linear measurements.
Subsection 2.1.1 Rectangular Coordinates in Three Dimensions
In CalcVR’s lesson on rectangular coordinates in three dimensions, the linear measurements of the coordinates are introduced. Several examples are given with the measurements of the coordinates displayed and a very brief description for each point. A description of the coordinates as a set of directions from the origin to a location is also given. This lesson includes with three questions that ask the user to select a point with a given set of coordinates.
This lesson continues with a description of rectangular coordinates in terms of a wireframe coordinate grid and octants. There are several follow-up questions involving the octants of several points and assessing which (if any) of the coordinates are fixed along different paths in space.
Activity 2.1.1. Drawing Points in 3D.
In this problem to move “forward” or “backwards” is to move in the direction of positive x (the x coordinate increases, the y and z remain the same) or negative x (the x coordinate decreases, the y and z remain the same), respectively; “to the right” or “to the left” is moving in the positive or negative y direction, respectively; “up” or “down” is moving in the positive or negative z direction, respectively:
(a)
Find the coordinates of the point \(A\) where one ends if one starts at point \((1,2,3)\) and moves 5 units forward, 4 units to the left, and 2 units up.
(b)
Draw the point \(A\) (that is your answer to task 2.1.1.a) on a set of 3D axes and include the line segments that show that coordinate’s points as a set of directions from the origin (like in the VR lesson).
(c)
Find the coordinates of the point \(B\) where one ends if one starts at point \((3,-4,2)\) and moves 4 units backwards, 4 units to the right, and 4 units down.
(d)
Draw the point \(B\) (that is your answer to task 2.1.1.c) on a set of 3D axes and include the line segments that show that coordinate’s points as a set of directions from the origin (like in the VR lesson).
Subsection 2.1.2 Comments to Instructors
The worksheet on Drawing Points in 3D should be done after the CalcVR lesson on Rectangular Coordinates in 3D. This worksheet is designed to help students take what they are doing in the virtual environment and apply it to their written (pencil and paper) work. We have made this into a worksheet so that students can click on the US or A4 buttons and be taken to page where they can fill in blanks and print thier work to a single page (or pdf file) to submit thier work for feedback.
