1. **Plotting points A-H in 3D space:** - Points: A(0,0,2), B(0,2,0), C(2,0,0), D(2,3,0), E(3,2,4), F(-2,0,4), G(-1,-1,-1), H(2,1,-2). - These points are located in the 3D coordinate system with axes x, y, and z. 2. **Cube with opposite corners at (0,0,0) and (4,4,4):** - The cube's vertices are all points where each coordinate is either 0 or 4. - Coordinates of other corners: (0,0,0), (4,0,0), (0,4,0), (0,0,4), (4,4,0), (4,0,4), (0,4,4), (4,4,4). 3. **Box with vertices at (0,0,0), (3,0,0), (0,2,0), (0,0,2):** - These are three edges meeting at the origin. - Other vertices are found by adding these vectors pairwise and all three: (3,2,0), (3,0,2), (0,2,2), (3,2,2). 4. **Rectangular parallelepiped with diagonal ends (0,0,0) and (4,5,3):** - Vertices are all combinations of coordinates 0 or the diagonal end: (0,0,0), (4,0,0), (0,5,0), (0,0,3), (4,5,0), (4,0,3), (0,5,3), (4,5,3). 5-7. **Planes:** - $x=0$: yz-plane. - $y=0$: xz-plane. - $z=0$: xy-plane. 8-9. **Planes parallel to xy-plane:** - $z=5$ and $z=-5$ are planes parallel to xy-plane at heights 5 and -5. 10. **Plane $x + y = 4$:** - A plane slanting through x and y axes. 11-16. **Planes of form $ax + by + cz = d$:** - 11: $3x + z = 12$ - 12: $2y + z = 6$ - 13: $x + z = 0$ - 14: $2x - y = 0$ - 15: $3y - z = 6$ - 16: $z - 4x = 8$ - Each represents a plane in 3D. 17-18. **Cylinders:** - 17: $x^2 + y^2 = 4$ is a cylinder along z-axis with radius 2. - 18: $(y-2)^2 + z^2 = 1$ is a cylinder along x-axis. 19-21. **Parabolic cylinders:** - 19: $x^2 = 9z$ opens along z. - 20: $y^2 = 4z$ opens along z. - 21: $(x-2)^2 = 8y$ opens along y. 22. **Ellipse:** - $4x^2 + 9y^2 = 36$ is an ellipse in xy-plane. 23-24. **Surfaces from quadratic forms:** - 23: $x^2 + z^2 - 4x - 6y + 9 = 0$ can be rewritten to identify shape. - 24: $x^2 + 4y^2 - 4x - 32y = 64$ similarly. **Summary:** - The first four problems involve plotting points and shapes in 3D. - Problems 5-24 describe various planes, cylinders, parabolas, ellipses, and quadratic surfaces.