Physical vector

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Corresponding Wikipedia article: Vector

The physical vector is a physical quantity at a physical space point, which is expressed as a projection of this vector on each geometric ray from this point. The maximal projection value is a magnitude (absolute value) of the vector, and the ray of this projection is a direction of the vector.

The vectors sum at a point is the sum of their projections on all geometric rays from the given point.

The angle between two vectors is a plane angle, which is determined by the solution of the vector triangle constructed by the infinitesimally small segments of the vector directional rays.

In a space of any dimensionality (>1), any two vectors with the known magnitudes and angle can be represented by the numbers on a complex plane, which is determined by the vector triangle. The basic operations on them are reduced to the simple operations with complex numbers:

Operation Notation Calculation
Addition - subtraction \[\mathbf{\overrightarrow{A}}\pm\mathbf{\overrightarrow{B}}\] \(A\pm B,\)
result is in the same plane
Dot product \[\mathbf{\overrightarrow{A}}\cdot\mathbf{\overrightarrow{B}}\] \(Re(A\cdot\overline{B})\)
Cross product \[\mathbf{\overrightarrow{A}}\times\mathbf{\overrightarrow{B}}\] Magnitude \(=Im(A\cdot\overline{B})\),
direction is perpendicular to the plane

The projection of a vector on a line (ray) is the dot product of this vector with a unit vector of this line (ray). The unit vector magnitude is the unit of physical quantity.

For a space of integer dimensionality N, as it is known, the N orthogonal projections exactly determine a vector. Such mathematical method is also known in the spectral transform (Fourier etc.), but unlike the traditional vector notion, the spectrum can be either discrete or continuous:

Spectral analysis Vector analysis
Basis Orthogonal functions. Orthogonal vectors (axes)
(special case)
Sequence of numbers (Fourier etc.), representation of a periodic function. Mathematical vector (N-tuple), physical vector of an integer dimensionality (N projections).
(general case)
Spectral density of an arbitrary function. Arbitrary physical vector, a set of projections.

Analytic geometry with its algebra of matrix, tensor etc. is applicable only to the special cases, but not to express the fundamental laws.

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