Distance and midpoints (3D) - AI/AA SL 3.1
Keywords
Distance in 3D (3차원에서의 거리), Midpoints in 3D (3차원에서의 중점), Three-dimensional space (3차원 공간), Applet usage (애플릿 사용), Midpoint calculation (중점 계산)
| Factual Questions | Conceptual Questions | Debatable Questions |
| What is the formula used to calculate the distance between two points in a 3D space? | Why is the concept of the midpoint important in understanding the geometry of 3D space? | To what extent are the mathematical concepts of distance and midpoints critical to advancements in 3D modeling and design? |
| How does one determine the coordinates of the midpoint between two points in 3D? | How does the understanding of distance and midpoints in 3D contribute to the field of vector analysis? | Can the reliance on digital tools for calculating distances and midpoints undermine the fundamental understanding of spatial relationships? |
| What changes occur to the midpoint when one point is held constant and the other is moved along one axis? | In what ways do the principles of distance and midpoint calculation extend to higher dimensions? | How might the interpretation of distance and midpoints differ in theoretical mathematics versus applied fields like engineering or physics? |
Exploring Distance and Midpoints in 3D
Mini-Investigation: Exploring Distance and Midpoints in 3D
Objective:
Understand the concepts of calculating distances and midpoints between points in a three-dimensional space using a given applet.
Questions:
1. What is the distance between point A at (1, 2, 3) and point B at (4, 5, 6) according to the applet?
2. How does the applet determine the midpoint between these two points?
3. If we move point A to (2, 3, 4), what happens to the midpoint? Calculate the new midpoint.
4. Experiment with the applet by setting point A at the origin (0, 0, 0). What do you observe about the midpoint as you move point B around?
5. How does the distance change if only one coordinate of point B increases, keeping the other two constant?
6. What geometric shape is formed when you trace the midpoint while moving point B in a straight line away from point A? Can you visualize it using the applet?
7. If the distance between two points is doubled, by what factor does the midpoint's coordinates change? Test your hypothesis using the applet.
8. Challenge: Can you find two points such that the distance between them is equal to the sum of their midpoints' coordinates?
Extension Activity:
Use the applet to find the distance and midpoint for the following sets of points:
- Points C (7, -2, 5) and D (-3, 4, -6)
- Points E (0, 0, 0) and F (10, 10, 10)
Reflect on your findings and discuss how understanding these concepts is crucial for fields such as engineering, computer graphics, and physics.
Part 2 - Checking for understanding
Watch the video below to see more and crystallize your findings.
