Vectors

[last updated: 2024-03-19
vector mathematics
vectors: alternative definitions
quaternions
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      On This Page:
• Description/Context
• Definitions
• Vector mathematical symbol
• Notation:
  "tuple" vs. "component" vs. matrix
• Equality
• Links/refs:

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• Description/Context:
  • Mathematics uses numbers to describe real-world values.
    Some real-world things only require a single number to describe them: mass, density, temperature, etc.
    But some things require two or more numbers to describe them.
    for example:
    • force and velocity require both magnitude and direction.
    • position on a graph (2-dim) or in space (3-dim) require x & y, or x, y, and z coordinates.
    • electrical impedance requires both magnitude and phase angle (if using polar coordinates), or resistance and reactance (if using Cartesian coordinates).

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• Definitions:
  Vectors are used to describe things that require more than one number to fully specify them, like the examples above.
  Vectors are lists of (at least two) numbers.

  The Dimension of a vector is the number of elements in its list.

  Scalars are quantities that only require one number to fully specify them.

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• Symbol:
  A common notation to identify a quantity as a vector is:
  Bold-face type with an arrow on top, sometimes with subscripts:
  A⃗ or B⃗ etc., or A⃗₁ or A⃗₂

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• Vector Notation: (tuples, components, and matrices)
  There are several different formats that can be used to represent/denote/specify/define a vector:
  • Tuples:
    • A⃗ = (a₁, a₂, a₃)     ["tuple" notation]
      a₁, a₂, and a₃ are the three "numbers" that define/specify the vector A⃗.
    • This format is commonly used to represent points in 3-d space, where a₁, a₂, and a₃ represent the x, y, and z coordinates of the point being defined.
    • In such usage, the vector is drawn as an arrow, with its starting point at the origin (= 0, 0, 0),
      and its tip at the point being defined.
    • Another example is Electrical Impedance:
      Z⃗ = (R, X)

      Impedance requires both Resistance (R) and Reactance (X) to fully specify it.
      It is plotted on a 2-d coordinate system, with R plotted in x axis, and X on the y.

      It can also be specified as: Z⃗ = (R, θ)
      with θ being the phase angle between voltage and current, and then plotted on a 2-d polar coordinate graph.

      Impedance is also usually drawn as an arrow, with its base at the origin and tip at the point being defined.

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  • Components:
    • A⃗ = a₁ i⃗ + a₂ j⃗ + a₃ k⃗       ["component" notation]
      Note this expression is an example of one of the operations that can be done on/between vectors,
      that being, "vector addition".
      Strictly speaking, a₁ i⃗, a₂ j⃗, and a₃ k⃗ are all vectors, and in this expression they're all added together to create the A⃗ vector.
      
    • For position vectors that represent points in space in a coordinate system,
      you can use algebraic or "component" notation using unit (or basis) vectors:
      Unit vectors are vectors of length 1, in the directions of the 3 Cartesian coordinate axes, x (i⃗), y (j⃗), and z (k⃗):
      i⃗ = (1, 0, 0)
      j⃗ = (0, 1, 0)
      k⃗ = (0, 0, 1)
      Properties of unit vectors: (see elsewhere for details of the "cross product" operator between vectors)
      i⃗ x j⃗ = k⃗
      j⃗ x k⃗ = i⃗
      k⃗ x i⃗ = j⃗
      These relationships mean that the vectors are all orthogonal, ie. mutually perpendicular.
      [In systems other than Cartesian coordinate systems, it may not be necessary that the basis vectors be orthogonal...]

    • Electrical Impedance in component notation would be:
      Z⃗ = R + Xj⃗

      In the science of electronic components, it happens that reactance is created by both inductance and capacitance.
      And since inductive reactance results in a positive phase angle between voltage and current, the point is plotted "above the line" of the x-axis of the graph, that is, in a "+ j⃗" direction.
      Capacitive reactance on the other hand creates a negative phase angle, so the points are plotted "below the line" of the x axis, that is, in a "- j⃗" direction.
      Accordingly, the equations for Z are usually written:
      Z⃗ = R + X_L j⃗
      Z⃗ = R - X_C j⃗

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  • Matrices:
    • Vectors can be represented in matrix form as either a row vector or a column vector:
      A = [a1, a2, a3]
      A =
      | a1 |
      | a2 |
      | a3 |

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• Equality:
  • Note: the concept of equality is defined differently by different authors.
    The definition presented here is my nominal best guess at the moment for what makes the most sense:
    Two vectors are equal when their "components" are equal.
    This definition implies that two "equal vectors" can be located anywhere in space,
    and as long as they have the same direction (ie. are parallel), and the same length, then they are "equal" by this definition.

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  ... .,.. ...

• "vector mathematics" then becomes the operations needed to perform various geometrical manipulations on those points,
  or on the lines or surfaces constructed from the points, eg:
  moving a point from one place to another,
  rotating a line or surface in relation to the coordinate axes,
  moving or rotating the coordinate axes and representing the defined points in terms of the new axes,
  finding length and direction (angle to axes) of the vector,
  finding angles between two vectors,
  finding a vector perpendicular to a given vector,
  etc.

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• Geometric vector types:
  • A "free" vector: a line segment with specified length and direction
    (A "line segment" is an object that extends between two points)

    Free vectors are commonly specified by stating their "coordinates,"
    that is, the position in space (in relation to a pre-defined Cartesian coordinate system) of their end-point,
    with the assumption that their "tails" are located at the "origin", where x=y=z=0

  • A "line" vector: a line segment with length and direction and a point defined in space through which it must pass.
  • A "bound" vector": a line segment with a fixed position in space, ie. its tail not necessarily at the origin.

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• Continue with:
  vector mathematics
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• Links/Refs:
  • Griffiths, David J., Introduction to Electrodynamics, Prentice Hall
  • (link to:) Vectors - wikipedia
  • (link to:) Vector Dot Product - wikipedia