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) |
| Discrete (special case) |
Sequence of numbers (Fourier etc.), representation of a periodic function. | Mathematical vector (N-tuple), physical vector of an integer dimensionality (N projections). |
| Continuous (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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