Distance Formula
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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:
- $d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$
3 Dimensions
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:
- $d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2 + \paren {z_1 - z_2}^2}$
Proof
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.
Thus, by Pythagoras's Theorem:
- $d^2 = \size {x_1 - x_2}^2 + \size {y_1 - y_2}^2$
and the result follows.
$\blacksquare$
Sources
- 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: $10.1$: Distance $d$ between Two Points $\map {P_1} {x_1, y_1}$ and $\map {P_2} {x_2, y_2}$
- For a video presentation of the contents of this page, visit the Khan Academy.