# Distances/Vectors

In mathematics and physics, a **vector** is a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.

## Theoretical vectors[edit]

**Def.**

- "a quantity that has both magnitude and direction"
^{[1]} - "the signed difference between two points"
^{[2]}or - an "ordered tuple representing a directed quantity or the signed difference between two points"
^{[2]}

is called a **vector**.

## Unit vectors[edit]

**Notation:** let '"`UNIQ--postMath-00000001-QINU`"' denote a unit vector in the ith direction.

**Def.** a "vector with length 1"^{[3]} is called a **unit vector**

## Force vectors[edit]

A force vector is a force defined in two or more dimensions with a component vector in each dimension which may all be summed to equal the force vector. Similarly, the magnitude of each component vector, which is a scalar quantity, may be multiplied by the unit vector in that dimension to equal the component vector.

- '"`UNIQ--postMath-00000002-QINU`"'

where '"`UNIQ--postMath-00000003-QINU`"' is the magnitude of the force in the ith direction parallel to the x-axis.

## Triclinic coordinate systems[edit]

A triclinic coordinate system has coordinates of different lengths (a ≠ b ≠ c) along x, y, and z axes, respectively, with interaxial angles that are not 90°. The interaxial angles α, β, and γ vary such that (α ≠ β ≠ γ). These interaxial angles are α = y⋀z, β = z⋀x, and γ = x⋀y, where the symbol "⋀" means "angle between".

## Monoclinic coordinate systems[edit]

In a monoclinic coordinate system, a ≠ b ≠ c, and depending on setting α = β = 90° ≠ γ, α = γ = 90° ≠ β, α = 90° ≠ β ≠ γ, or α = β ≠ γ ≠ 90°.

## Orthorhomic coordinate systems[edit]

In an orthorhombic coordinate system α = β = γ = 90° and a ≠ b ≠ c.

## Tetragonal coordinate systems[edit]

A tetragonal coordinate system has α = β = γ = 90°, and a = b ≠ c.

## Rhombohedral coordinate systems[edit]

A rhombohedral system has a = b = c and α = β = γ < 120°, ≠ 90°.

## Hexagonal coordinate systems[edit]

A hexagonal system has a = b ≠ c and α = β = 90°, γ = 120°.

## Cubic coordinate systems[edit]

A cubic coordinate system has a = b = c and α = β = γ = 90°.

For two points in cubic space (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}), with a vector from point 1 to point 2, the distance between these two points is given by

- '"`UNIQ--postMath-00000004-QINU`"'

## Hypotheses[edit]

- For a vector, the direction can be stated and the magnitude is arbitrary.

*See also: Control groups, Proof of concept, and Proof of technology*

## See also[edit]

## References[edit]

- ↑
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^{2.0}^{2.1}**Lua error in Module:Citation/CS1 at line 3505: bad argument #1 to 'pairs' (table expected, got nil).** - ↑ "unit vector, In:
*Wiktionary*". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-08-10.

## External links[edit]

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