Main Menu

search

You are here

Home

Vector Mathematics:

[last updated: 2024-03-18]
vectors home page
-----


          On This Page:
    • Introduction
    • Vector Magnitude
    • Vector Operations - on and between vectors:
      • Vector addition
      • Scalar multiplication
      • Dot product (scalar product)
      • Cross product (vector product)
    • Vector translations:
      • rotations, etc.
    • Other things to study/develop...:
      • see also: matrices
      • handedness
      • del, curl, ... , other differentiation quantities
    • Links/refs:

---------------------------------------------------------

  • Introduction:
    "Doing mathematics" on and with vectors must start with these requirements:
            [these requirements are a consequence of the fact that vectors belong to a "vector space"]
    All vectors must have two operations defined, and those operations must in turn obey 8 axioms.
    • The operations are:
      vector addition, and
      scalar multiplication
    • The axioms are:
      • Vector addition is associative.
      • Vector addition is commutative.
      • Additive identity: ie. the existence of the additive "Zero" vector
      • Additive inverse: ie. the existence of the "negative" vector
      • "Compatibility of scalar multiplication with field multiplication" per wiki... (=== "Scalar multiplication is associative" to non-mathematicians...)
        a(bv) = (ab)v
      • Multiplicative identity: ie. the existence of the multiplicative "Unit" vector
      • Scalar multiplication is distributive across vector addition.
        a(u + v) = au + av
      • Scalar addition can be decomposed and distributed across scalar-vector multiplication.
        for mathematicians: "scalar multiplication is distributive with respect to field addition"
        ( (a+b) V = aV + bV )

    finally proceed to more complex operations, dot and cross products
    -------------------------------------------------------

  • Magnitude (scalar length) of a vector:
    denoted: |A⃗|
    • If a vector is defined as A⃗ = (a1, a2, a3),
      that is, a vector with starting point at the coordinate origin = (0, 0, 0)
      then the length will be:
      |A⃗| = √(a12 + a22 + a32)
    • If a vector is defined as starting at a point S = (s1, s2, s3)
      and ending at a point E = (e1, e2, e3)
      then its length is:
      |A⃗| = √( (e1-s1)2 + (e2-s2)2 + (e3-s3)2 )
      (see also the derivation in terms of dot product below)

    -------------------------------------------------------

  • Vector addition:
    • creates another vector
      if: A⃗ = (a, b, c)
      and: B⃗ = (d, e, f)
      then: C⃗ = A⃗ + B⃗ = ( a+d, b+e, c+f )

    • Addition is commutative:
        A⃗ + B⃗ = B⃗+ A⃗
    • Addition is associative:
        A⃗ + ( B⃗ + C⃗ ) = ( A⃗ + B⃗ ) + C⃗
    • Subtraction of vectors is addition of a negative vector:
        A⃗ - B⃗ = A⃗ + (-B⃗)
        -B⃗ is a vector of the same magnitude as B⃗, but in the opposite direction:
        if: A⃗ = (a, b, c)
        then: -A⃗ = (-a, -b, -c)

    -------------------------------------------------------

  • Scalar Multiplication:
    • creates another vector
    • multiplies the magnitude of the vector by the scalar amount.
      If: A⃗ = (a, b, c)
      and: k is a scalar,
      then: kA⃗ = (ka, kb, kc)
    • Direction is unchanged, but if the scalar is negative, the direction of the resulting product is reversed.
    • scalar multiplication is distributive:
      a ( A⃗ + B⃗ ) = aA⃗ + aB⃗

    -------------------------------------------------------

  • Dot product:   (Also called - Scalar product or Inner product, or Projection product)
    • creates a scalar value, ie. NOT a vector
      (the two vectors must be of the same degree, ie. the same number of dimensions/arguments)
    • Formally, for n dimensions:
        given: A⃗ = (a1, a2, ..., an)
        and: B⃗ = (b1, b2, ..., bn)
        and: n = number of dimensions/degree of A⃗ and B⃗
        then: A⃗ • B⃗ = ∑i=1-n ai bi = a1 b1 + a2 b2 + ... + an bn

      --------------------------

    • For two 3-d vectors:
        if: A⃗ = (a, b, c)
        and: B⃗ = (d, e, f)
        then: r = A⃗ • B⃗ = ad + be + cf

      --------------------------

    • alternatively:
        r = |A⃗| |B⃗| cos θ
            (where θ = angle between A⃗ and B⃗)
        • If A⃗ and B⃗ are parallel, then θ = 0, and cos θ = 1, giving:
          A⃗ • B⃗ = |A⃗| |B⃗|
        • if A⃗ and B⃗ are perpendicular, then θ = 90, and cos θ = 0, giving:
          A⃗ • B⃗ = 0

        Note however, that this definition requires that A and B be defined in 2-d space,
        or, if defined in 3-d space, that they be coplanar.
        That is, they must both pass through the origin (or any other common point).
        But if placed anywhere in "free" 3-d space, A and B may not be co-planar, so θ would not be defined.

      --------------------------

    • Dot product operation is commutative:
      A⃗ • B⃗ = B⃗ • A⃗
    • Dot product is distributive:
      A⃗ • ( B⃗ + C⃗ ) = A⃗ • B⃗ + A⃗ • C⃗
    • For any vector:
      A⃗ • A⃗ = |A⃗|2
      therefore:
      |A⃗| = √( A⃗ • A⃗ )
      --------------------------

    • Geometrically:
        (from Griffiths) "A dot B is the product of A times the projection of B along A (or vice versa)" ???

    -------------------------------------------------------

  • Cross-product:   (Also called - vector product)
    • creates another vector
    • if: A⃗ = (a1, a2, a3)
      and: B⃗ = (b1, b2, b3)
      A⃗ x B⃗ = C⃗ = ( (a2*b3 - b2*a3), (a3*b1 - b3*a1), (a1*b2 - a2*b1) )
    • |C⃗| = |A⃗| |B⃗| sin θ
      in a direction perpendicular to the plane created by A⃗ and B⃗,
      in a direction according to the right-hand rule:
        palm perpendicular to plane formed by the two vectors,
        fingers in direction of first vector, curling towards second vector (shortest direction),
        thumb points in direction of cross-product vector.
    • Cross product is distributive:
      A⃗ x (B⃗ + C⃗) = (A⃗ x B⃗) + (A⃗ x C⃗)
    • Cross product is "inverse commutative":
      A⃗ x B⃗ = - (B⃗ x A⃗)
    • if A⃗ and B⃗ are parallel, then:
      A⃗ x B⃗ = 0
      and in general:
      A⃗ x A⃗ = 0
    • If A and B are perpendicular, then sin θ = 1, so:
      A⃗ x B⃗ = 1
    • Geometrically:
      A⃗ x B⃗ results in a third vector in 3d space that is perpendicular to the plane created by the first two vectors.
      |A⃗ x B⃗| = area of parallelogram created by A⃗ and B⃗

    -------------------------------------------------------

  • Other things...:
    • Translations: moving or rotating coordinate axes.
    • other concepts to investigate: curl, gradient, divergence, flux, circulation, moment
    • Implicit in these definitions is the concept of direction, and especially the conventions of right-handedness or left-handedness.
      These concepts need a more rigorous development...
    • Del or nabla:

    -------------------------------------------------------

  • Links/Refs:

    -------------------------------------------------------

.

eof

Powered by Drupal