Distance Formula

Definition

The Distance Formula calculates the length of the line segment connecting two points in a Cartesian plane. For two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• How to read: “The distance d equals the square root of the quantity x sub two minus x sub one squared plus the quantity y sub two minus y one squared.”
• Meaning: The straight-line distance between two points, derived from the Pythagorean theorem.

Why It Matters

The distance formula is the “GPS” of analytic geometry, bridging abstract coordinates and physical distance. It is foundational for computer graphics, navigation systems, and physics simulations, allowing for the calculation of the shortest path or the magnitude of displacement.

Core Concepts

• Pythagorean Derivation: The formula is a direct application of the Pythagorean theorem ($a^2 + b^2 = c^2$) to a right triangle where the horizontal leg is $\Delta x = x_2 - x_1$ and the vertical leg is $\Delta y = y_2 - y_1$.
• Euclidean Norm: In linear algebra, this formula represents the $L_2$ norm (magnitude) of the displacement vector between the two points.
• Dimensional Generalization: The principle extends to three dimensions (and higher) by adding the squared difference of the additional coordinates: $d = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}$.
• Invariant Property: Distance is always non-negative ($d \ge 0$) and is zero if and only if the two points are identical.

Connected Concepts

• Midpoint Formula
• Pythagorean Theorem
• Distance Between Points in Space
• Analytic Proof Strategies
• Vectors in 2D