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 classical mechanicsEdit

Def. an "[amount of] intervening space between two points,[1] usually geographical points, usually (but not necessarily) measured along a straight line"[2] is called a distance.

Def. the "inevitable progression into the future with the passing of present events into the past"[3] or the "inevitable passing of events from future to present then past"[4] is called time.

Def. the "quantity of matter which a body contains, irrespective of its bulk or volume"[5] or a "quantity of matter cohering together so as to make one body, or an aggregation of particles or things which collectively make one body or quantity"[6] is called a mass.

Def. the "rate of motion or action, specifically[7] the magnitude of the velocity;[8] the rate distance is traversed in a given time"[9] is called the speed.

Theoretical vectorsEdit

Def.

  1. "a quantity that has both magnitude and direction"[10]
  2. "the signed difference between two points"[11] or
  3. an "ordered tuple representing a directed quantity or the signed difference between two points"[11]

is called a vector.

Unit vectorsEdit

Notation: let   denote a unit vector in the ith direction.

Def. a "vector with length 1"[12] is called a unit vector

Force vectorsEdit

 
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.

 

where   is the magnitude of the force in the ith direction parallel to the x-axis.

Triclinic coordinate systemsEdit

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 systemsEdit

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 systemsEdit

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

Tetragonal coordinate systemsEdit

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

Rhombohedral coordinate systemsEdit

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

Hexagonal coordinate systemsEdit

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

Cubic coordinate systemsEdit

 
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

 

HypothesesEdit

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

See alsoEdit

ReferencesEdit

  1. Emperorbma (17 August 2003). "distance". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 24 May 2019.
  2. Brya (17 January 2006). "distance". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 24 May 2019.
  3. DAVilla (3 January 2009). "time". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 23 July 2019.
  4. 24.13.132.38 (23 September 2005). "time". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 23 July 2019.
  5. Eclecticology (12 September 2003). "mass". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2013-08-12.
  6. Emperorbma (14 November 2003). "mass". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2014-02-28.
  7. Widsith (15 May 2006). "speed". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 4 January 2020.
  8. Connel MacKenzie (23 November 2005). "speed". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 4 January 2020.
  9. Emperorbma (14 November 2003). "speed". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 4 January 2020.
  10. Paul G (22 December 2003). "vector". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-08-10.
  11. 11.0 11.1 "vector". San Francisco, California: Wikimedia Foundation, Inc. 24 July 2015. Retrieved 2015-08-10.
  12. Language Lover (1 February 2007). "unit vector". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-08-10.

External linksEdit