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

["# Understanding the Distance Between Two Points in 3D Space: A Clear Look at √2", "When calculating distances in three-dimensional (3D) space, geometry becomes both fascinating and precise. If you’ve ever wondered how to find the straight-line distance between two points not on the same axis—like the points ( (1,1,0) ) and ( (1,0,1) )—this article breaks down the formula and computation step by step.", "## What Is the Distance Between Two Points in 3D?", "In three-dimensional Cartesian coordinates, the distance ( d ) between any two points ( (x_1, y_1, z_1) ) and ( (x_2, y_2, z_2) ) is given by the Euclidean distance formula:", "[\nd = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\n]", "This formula extends the familiar 2D distance concept into three dimensions, accounting for differences along the ( x ), ( y ), and ( z ) axes separately.", "## Applying the Formula to Points ( (1,1,0) ) and ( (1,0,1) )", "Let’s plug the coordinates into the distance formula:", "- Point A: ( (x_1, y_1, z_1) = (1,1,0) )\n- Point B: ( (x_2, y_2, z_2) = (1,0,1) )", "Now calculate each coordinate difference:", "- ( x_2 - x_1 = 1 - 1 = 0 )\n- ( y_2 - y_1 = 0 - 1 = -1 )\n- ( z_2 - z_1 = 1 - 0 = 1 )", "Next, square each difference:", "- ( (0)^2 = 0 )\n- ( (-1)^2 = 1 )\n- ( (1)^2 = 1 )", "Add the squared differences:", "[\n0 + 1 + 1 = 2\n]", "Finally, take the square root of the sum:", "[\n\sqrt{2}\n]", "## Conclusion", "The distance between the points ( (1,1,0) ) and ( (1,0,1) ) in 3D space is exactly ( \sqrt{2} ), approximately 1.414 units. This simple example illustrates how 3D distance calculations rely on summing the squared differences across all coordinate axes and applying the square root of the total sum—showcasing a clean and reliable method used in physics, computer graphics, engineering, and many other fields.", "Understanding this formula opens the door to analyzing spatial relationships with clarity and precision!"]