Distance Formula/3 Dimensions

Theorem

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:

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

Hence $d$ is the length of the straight line segment $AB$.

Proof

Let $d$ be the distance to be found between $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$.

Let the points $C$ and $D$ be defined as:

$C = \tuple {x_2, y_1, z_1}$

$D = \tuple {x_2, y_2, z_1}$

Let $d'$ be the distance between $A$ and $D$.

From Distance Formula, it can be seen that:

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

We note that $\triangle ADB$ is a right triangle.

Thus by Pythagoras's Theorem:

$AB^2 = AD^2 + DB^2$

Thus:

and so:

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

as it was to be proved.

$\blacksquare$

Sources

• 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
• 1947: William H. McCrea: Analytical Geometry of Three Dimensions (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $2$. Cartesian Coordinates
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 12$: Formulas from Solid Analytic Geometry: Distance $d$ between Two Points $\map {P_1} {x_1, y_1, z_1}$ and $\map {P_2} {x_2, y_2, z_2}$: $12.1$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 10$: Formulas from Solid Analytic Geometry: Distance $d$ between Two Points $\map {P_1} {x_1, y_1, z_1}$ and $\map {P_2} {x_2, y_2, z_2}$: $10.1$