91Ó°ÊÓ

When working with 3D coordinates, we are extending the traditional 2D coordinate system into a three-dimensional space. This means we have an extra axis to consider. In a 3D system, every point is defined by three numbers:

• x (the horizontal position),
• y (the vertical position), and
• z (the depth position).

These three coordinates give us a unique point in space. For example, in our exercise, we have points (4, 1, 5) and (8, 2, 6). Each of these points represents a location in a 3D space, characterized by values of x, y, and z respectively. The addition of the z-coordinate allows us to handle problems involving more depth and complexity compared to 2D systems.
Visualizing 3D points can be challenging, but imagine them as lying on a grid, much like a stack of 2D planes, with z describing how 'tall' from the base plane each point is.

Calculating the distance between two points in a 3D space involves expanding on the Pythagorean theorem from 2 dimensions to 3 dimensions. In 2D, the distance between two points years would calculate using \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\]however, in 3D space, we include the third dimension. This means we add the difference in z-coordinates as well:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.\]This formula might look complex, but it simply accounts for the spatial distance along the extra z-axis. For our example points (4, 1, 5) and (8, 2, 6), we did:

• x-difference: 8 - 4 = 4
• y-difference: 2 - 1 = 1
• z-difference: 6 - 5 = 1

Then square each, sum them up, and finally take the square root of the total to get the distance.

The concept of squared differences is crucial for calculating distances, particularly in higher dimensions. This step ensures that every coordinate's contribution to the overall distance is always positive. You cannot have a 'negative' distance, so squaring accounts for any directional differences.

• A difference like (8 - 4) becomes (4)^2 = 16.
• (2 - 1) becomes (1)^2 = 1.
• And similarly, (6 - 5) leads to (1)^2 = 1.

Square each difference, and you ensure they all effectively measure 'how far apart' the values are, without direction bothering them.
This approach is why even if one coordinate value is lesser at the second point, we don’t end up with a negative distance. Adding all these squared distances, in our example 16, 1, and 1, results in 18, which is then used to find the actual distance by taking a square root in the following step.