Distance Formula

Theorem

The distance $d$ between two points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ on a Cartesian plane is:

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

The distance $d$ between two points $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$ in a Cartesian space of 3 dimensions is:

$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2 + \paren {z_1 - z_2}^2}$

The distance in the horizontal direction between $A$ and $B$ is given by $\size {x_1 - x_2}$.

The distance in the vertical direction between $A$ and $B$ is given by $\size {y_1 - y_2}$.

By definition, the angle between a horizontal and a vertical line is a right angle.

So when we place a point $C = \tuple {x_1, y_2}$, $\triangle ABC$ is a right triangle.

$d^2 = \size {x_1 - x_2}^2 + \size {y_1 - y_2}^2$

and the result follows.

$\blacksquare$

1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $4$. Distance between two points
1936: Richard Courant: Differential and Integral Calculus: Volume $\text { II }$ ... (previous) ... (next): Chapter $\text I$: Preliminary Remarks on Analytical Geometry and Vector Analysis: $1$. Rectangular Co-ordinates and Vectors: $1$. Coordinate Axes
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (next): $\text {III}$. Analytical Geometry: The Straight Line
1958: P.J. Hilton: Differential Calculus ... (previous) ... (next): Chapter $1$: Introduction to Coordinate Geometry: $(1.1)$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: Distance $d$ between Two Points $\map {P_1} {x_1, y_1}$ and $\map {P_2} {x_2, y_2}$: $10.1$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 8$: Formulas from Plane Analytic Geometry: Distance $d$ between Two Points $\map {P_1} {x_1, y_1}$ and $\map {P_2} {x_2, y_2}$: $8.1.$