Thus, the fourth vertex is $\boxed(0,1,1)$. - Coaching Toolbox

Understanding the Context

In Cartesian 3D space, every point is defined by three coordinates: $x$, $y$, and $z$. The vertex $oxed{(0,1,1)}$ occupies a unique position: its $x$-coordinate is 0, while $y$ and $z$ are both 1. This places the point precisely halfway between the positive $y$-axis and $z$-axis, elevated slightly in both directions from the origin. Its location embodies symmetry and utility, offering a foundational anchor in frameworks involving planes, edges, and faces.

Critical for Solid Construction

When forming polyhedra—such as tetrahedrons—the placement of each vertex determines the shape’s volume, angles, and surface properties. The inclusion of $oxed{(0,1,1)}$ often completes a minimal set of points required for convex hulls in 3D. Combined with the origin $oxed{(0,0,0)$, edge endpoints like $oxed{(1,0,0)}$ and $oxed{(0,1,0)}$, this vertex contributes to defining a flat triangular base or an angular edge structure. Together, these points yield faces with predictable orientation and spatial relationships—essential for mathematical modeling, computer graphics, and engineering applications.

1. Computer Graphics & Game Design:
   In 3D rendering, vertex coordinates dictate how objects are rendered and interact. The precise positioning of $oxed{(0,1,1)}$ ensures smooth faceting and accurate shading, contributing to realistic or stylized visual representations.

2. Mathematical Proofs & Theorems:
   Geometric proofs often rely on coordinate-based verification. The point $oxed{(0,1,1)}$ frequently appears in coordinate geometry problems, helping demonstrate coplanarity, tetrahedral volume, or distance relationships.

3. Robotics & Spatial Planning:**
   In automation and spatial mapping, defined vertex positions guide robotic pathfinding and 3D environment modeling. The placement of $oxed{(0,1,1)}$ supports efficient navigation algorithms and collision avoidance systems.

Why This Vertex Stands Out

Among potential coordinates in a 3D grid, $oxed{(0,1,1)}$ offers both simplicity and utility. Its location avoids symmetry overload while providing key directional balance, ideal as a “corner” in non-axis-aligned constructions. With unit values across $y$ and $z$, it serves as a modular building block adaptable to expansions and variations.

Final Thoughts

Conclusion

The vertex $oxed{(0,1,1)}$ exemplifies how a single point in 3D space can profoundly influence geometric integrity and functional outcomes. Whether embedded in theoretical problems, visual simulations, or spatial systems, this coordinate anchors a network of spatial logic—making it indispensable in both academic study and practical innovation.

Next time you analyze a tetrahedron or plot points in 3D space, recognize the subtle yet vital role of this key vertex: $oxed{(0,1,1)}$—a testament to precision in geometry.