# 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

Def.

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

is called a vector.

## Unit vectors

Notation: let ${\displaystyle {\hat {i}}}$ denote a unit vector in the ith direction.

Def. a "vector with length 1"[3] is called a unit vector

## Force vectors

The diagram breaks down a force vector relative to coordinate axes x and y. Credit: HUB.

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.

${\displaystyle {\vec {F}}={\vec {F}}_{x}+{\vec {F}}_{y}+{\vec {F}}_{z}=F_{x}{\hat {i}}+F_{y}{\hat {j}}+F_{z}{\hat {k}},}$

where ${\displaystyle F_{x}}$ is the magnitude of the force in the ith direction parallel to the x-axis.

## Triclinic coordinate systems

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

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

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

## Tetragonal coordinate systems

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

## Rhombohedral coordinate systems

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

## Hexagonal coordinate systems

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

## Cubic coordinate systems

A Hexahedron is a cube; a regular polyhedron. Credit: Kjell André.

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

For two points in cubic space (x1, y1, z1) and (x2, y2, z2), with a vector from point 1 to point 2, the distance between these two points is given by

${\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}.}$

## Hypotheses

Main source: Hypotheses
1. For a vector, the direction can be stated and the magnitude is arbitrary.