Distance in 3D (3차원에서의 거리), Midpoints in 3D (3차원에서의 중점), Three-dimensional space (3차원 공간), Applet usage (애플릿 사용), Midpoint calculation (중점 계산)

[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]What is the formula used to calculate the distance between two points in a 3D space?[/td][br][td]Why is the concept of the midpoint important in understanding the geometry of 3D space?[/td][br][td]To what extent are the mathematical concepts of distance and midpoints critical to advancements in 3D modeling and design?[/td][br][/tr][br][tr][br][td]How does one determine the coordinates of the midpoint between two points in 3D?[/td][br][td]How does the understanding of distance and midpoints in 3D contribute to the field of vector analysis?[/td][br][td]Can the reliance on digital tools for calculating distances and midpoints undermine the fundamental understanding of spatial relationships?[/td][br][/tr][br][tr][br][td]What changes occur to the midpoint when one point is held constant and the other is moved along one axis?[/td][br][td]In what ways do the principles of distance and midpoint calculation extend to higher dimensions?[/td][br][td]How might the interpretation of distance and midpoints differ in theoretical mathematics versus applied fields like engineering or physics?[/td][br][/tr][br][/table][br]

Mini-Investigation: Exploring Distance and Midpoints in 3D[br][br]Objective:[br]Understand the concepts of calculating distances and midpoints between points in a three-dimensional space using a given applet.[br][br]Questions:[br]1. What is the distance between point A at (1, 2, 3) and point B at (4, 5, 6) according to the applet?[br]2. How does the applet determine the midpoint between these two points?[br]3. If we move point A to (2, 3, 4), what happens to the midpoint? Calculate the new midpoint.[br]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?[br]5. How does the distance change if only one coordinate of point B increases, keeping the other two constant?[br]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?[br]7. If the distance between two points is doubled, by what factor does the midpoint's coordinates change? Test your hypothesis using the applet.[br]8. Challenge: Can you find two points such that the distance between them is equal to the sum of their midpoints' coordinates?[br][br]Extension Activity:[br]Use the applet to find the distance and midpoint for the following sets of points:[br]- Points C (7, -2, 5) and D (-3, 4, -6)[br]- Points E (0, 0, 0) and F (10, 10, 10)[br][br]Reflect on your findings and discuss how understanding these concepts is crucial for fields such as engineering, computer graphics, and physics.[br]

Watch the video below to see more and crystallize your findings.