Distance

Standard Definition: Distance is a scalar measure of the interval between two locations, determined by their absolute difference. In a Euclidean space, the distance between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is given by:

For three-dimensional space, the formula extends to account for the ( z ) coordinate as well.

ELI5 with Real-life Example: Imagine you're at a park, and you see two trees. You want to know how far apart these trees are. You could use a measuring tape to find out, but let's say you know how far each tree is from a particular spot (like a bench) along both east-west and north-south directions.

If one tree is 3 meters to the east and 4 meters to the north of the bench, and the other tree is right next to the bench, the "distance" between the two trees is the length of the straight line you'd walk from one tree to the other. Using the Pythagoras theorem (like in right-angled triangles), you'd find this straight path is 5 meters long.

In everyday life, distance helps us understand how far apart two things are. Whether it's finding out how far your home is from school or determining the length of a football field, the concept of distance is fundamental.

Here's the visualization of the distance concept using the example of trees in a park:

• The blue path represents the eastward movement from the bench, a distance of 3 meters.
• The green path represents the northward movement from the end of the blue path, covering 4 meters.
• The red path is the direct path (or the distance) between the bench and the tree, which is 5 meters.

You can think of this as a journey where you first move eastward, then northward, and realize that the direct path (or the "shortcut") between your starting and ending points is the diagonal red line. This visualization helps connect the concept of distance with the Pythagorean theorem in a 2D space.