\text{Distance between } (0,0,0) \text{ and } (1,1,0): \sqrt{(1)^2 + (1)^2 + 0} = \sqrt{2} \\

["# Understanding the Distance Between Points in 3D Space: The Case of (0,0,0) and (1,1,0)", "When working with 3D coordinate systems, calculating the distance between two points is a foundational concept with broad applications in geometry, physics, engineering, and computer graphics. A classic example involves finding the distance between points ( (0,0,0) ) and ( (1,1,0) ). This simple yet powerful calculation introduces key principles behind the distance formula in three-dimensional space.", "## The 3D Distance Formula Explained", "The general formula to compute the distance ( d ) between two points ( (x_1, y_1, z_1) ) and ( (x_2, y_2, z_2) ) is derived from the Pythagorean theorem:", "[\nd = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\n]", "Applying this formula to the points ( (0, 0, 0) ) and ( (1, 1, 0) ):", "[\nd = \sqrt{(1 - 0)^2 + (1 - 0)^2 + (0 - 0)^2} = \sqrt{1^2 + 1^2 + 0^2} = \sqrt{1 + 1 + 0} = \sqrt{2}\n]", "So, the distance between these two points is exactly ( \sqrt{2} ).", "## Why This Formula Works", "Imagine a right-angled triangle (or better, a right-angled tetrahedron) formed in 3D space, where each edge corresponds to the differences in x, y, and z coordinates. The line segment connecting the points acts as the hypotenuse, and the squared differences represent squared leg lengths. Applying the Pythagorean theorem twice — first in the xy-plane, then extend it — leads naturally to the 3D distance formula.", "## Real-World Applications", "Understanding this distance helps in numerous fields:", "- Physics: Calculating distances between particles or objects in 3D environments.\n- Computer Graphics: Determining spatial relationships for rendering scenes.\n- Robotics: Programming precise movement paths.\n- Navigation and Mapping: GPS-based route calculations.", "## Conclusion", "The distance formula in three dimensions is a simple yet powerful tool rooted in geometric principles. The precise calculation of the distance between ( (0,0,0) ) and ( (1,1,0) ) as ( \sqrt{2} ) exemplifies how coordinate systems model real-world space. Mastering this concept simplifies more complex spatial reasoning and lays groundwork for advanced topics in math and applied sciences.", "---", "Keywords: 3D distance formula, distance between points in 3D, Euclidean distance 3D, coordinate geometry, spatial distance, geometry fundamentals, math explanation."]