Power vectors versus power matrices, and the mathematical nature of dioptric power

Vectors are mathematical entities characterized by both magnitude and direction. Unlike scalars, which are defined solely by magnitude, vectors represent quantities like displacement, velocity, and force, where direction is essential. Vectors are graphically represented as directed line segments, extending from an initial point to a terminal point, denoted with bold letters or arrows placed above the symbol. Two vectors are deemed equal if they share identical magnitudes and directions,...

The power factor is defined as the ratio of average (or active) power to apparent power, as illustrated by the relation

The dot product, or scalar product, is a fundamental operation in vector algebra that combines two vectors to yield a scalar quantity. It is particularly valuable in physical applications, such as calculating work, and in mathematical contexts, such as determining vector projections and direction cosines.For vectors a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩, the dot product is defined as:\begin{equation*}\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\end{equation*}This operation is...

Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of the angle between both the vectors and the direction perpendicular to both the individual vectors. As there are always two directions perpendicular to a given plane, one on each side, the direction of the vector product is governed by the right-hand thumb rule.
Consider the cross product of two vectors. Imagine rotating the first vector about...

Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of the angle between both the vectors and the direction perpendicular to both the individual vectors. As there are always two directions perpendicular to a given plane, one on each side, the direction of the vector product is governed by the right-hand thumb rule.
Consider the cross product of two vectors. Imagine rotating the first vector about...

Power engineers have introduced the concept of complex power to determine the cumulative effect of parallel loads. This idea plays a crucial role in power analysis because it encompasses all the details related to the power consumed by a specific load.
Complex power is defined as the multiplication of the voltage and the complex conjugate of the current. The magnitude of this power, known as apparent power, is measured in volt-amperes (VA). Notably, the angle of the complex power equates to the...