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Problem: Plot the following points in three-dimensional space:

• $P(0, 0, 2)$
• $Q(0, 2, 2)$
• $R(2, 1, 2)$
• $S(-2, -1, 2)$

These points have coordinates in 3D space, and all share the same $z$-coordinate, which is $2$. This means they lie on the same plane $z = 2$.

Here’s a breakdown of how these points would appear:

• $P(0,0,2)$: This point lies on the z-axis at $z=2$.
• $Q(0,2,2)$: This point is 2 units along the positive y-axis at $z=2$.
• $R(2,1,2)$: This point is located 2 units along the x-axis and 1 unit along the y-axis at $z=2$.
• $S(-2,-1,2)$: This point is 2 units along the negative x-axis and 1 unit along the negative y-axis at $z=2$.

Would you like me to help visualize or explain more about the structure of these points in 3D?

Here are 5 questions to deepen understanding:

1. How can we determine the distance between two points in 3D space?
2. What is the importance of plotting points in three dimensions versus two dimensions?
3. What geometric shape might these points suggest when connected?
4. How does the plane $z = 2$ influence the positioning of these points?
5. How can we calculate the midpoint between any two of these points?

Tip: In three-dimensional plots, the order of the coordinates (x, y, z) plays a crucial role, and misplacing them can distort the graph. Always verify coordinates carefully.