Definition:Linear Measure/Length

Definition

Length is linear measure taken in a particular direction.

Usually, in multi-dimensional figures, the dimension in which the linear measure is greatest is referred to as length.

It is the most widely used term for linear measure, as it is the standard term used when only one dimension is under consideration.

Length is the fundamental notion of Euclidean geometry, never defined but regarded as an intuitive concept at the basis of every geometrical theorem.

Vector Definition

Let $A$ and $B$ be points in a real Euclidean space $\R^n$.

Let $\mathbf a$ and $\mathbf b$ be the position vectors of $A$ and $B$ respectively.

The length of the straight line $AB$ is given by:

$\map L {AB} := \size {\mathbf a - \mathbf b}$

where:

$\mathbf a - \mathbf b$ denotes the vector subtraction operation

$\size {\mathbf a - \mathbf b}$ denotes the magnitude of $\mathbf a - \mathbf b$.

Also see

• Definition:Breadth
• Definition:Depth
• Definition:Height (Linear Measure)

Sources

• 1947: William H. McCrea: Analytical Geometry of Three Dimensions (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $\S 1$. Introductory: Nomenclature
• 1952: T. Ewan Faulkner: Projective Geometry (2nd ed.) ... (previous) ... (next): Chapter $1$: Introduction: The Propositions of Incidence: $1.1$: Historical Note
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): length