Distances/Vectors

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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.

  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[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]

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.

'"`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]

Reseaux 3D aP.png

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]

Monoclinic.png

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]

Reseaux 3D oP.png

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

Tetragonal coordinate systems[edit]

Reseaux 3D tP-2011-03-12.png

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

Rhombohedral coordinate systems[edit]

Rhombohedral.svg

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

Hexagonal coordinate systems[edit]

Reseaux 3D hP.png

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

Cubic coordinate systems[edit]

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

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

Hypotheses[edit]

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

See also[edit]

References[edit]

  1. Lua error in Module:Citation/CS1 at line 3505: bad argument #1 to 'pairs' (table expected, got nil).
  2. 2.0 2.1 Lua error in Module:Citation/CS1 at line 3505: bad argument #1 to 'pairs' (table expected, got nil).
  3. "unit vector, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Retrieved 2015-08-10. 

External links[edit]

{{Physics resources}}

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