Vector methods

Vector methods (physics)

Methods that make use of the behavior of physical quantities under coordinate transformations.

From the point of view of physics, the most appropriate definition of a vector in three-dimensional space is a quantity that has three components which transform under rotations of the coordinate system like the coordinates of a point in space. What characterizes rotations is that the distance from the origin—

of all points x with cartesian coordinates x_i, i = 1, 2, 3—remains unchanged. Specifically, if the rotation takes the x_i to new coordinates x_i^′, given by Eqs. (1) [in which det{a_ij} is

(1) 

the determinant of the matrix {a_ij}], then the three quantities V_i, i = 1, 2, 3, form the components of a vector V , if in the new coordinate system the transformed coordinates are given by Eq. (2).

(2) 

If the coordinate transformation of Eqs. (1) is such that for i = j and a_ij = 0 for i ≠ j, but such that det{a_ij} = -1 rather than +1 as in Eqs. (1), then it describes a reflection, in which a right-handed coordinate system is replaced by a left-handed one. If, for such a transformation, Eq. (2) also holds, then V is called a polar vector, whereas if the components of V do not change sign, it is called an axial vector or pseudovector. The vector V can also be looked upon as a quantity with a direction, with the magnitude pointing from the origin of the coordinate system to the point in space with the cartesian coordinates (V₁, V₂, V₃).

A quantity that remains invariant under a rotation of the coordinate system is called a scalar. The importance of vectors and scalars in physics derives from the assumed isotropy of the universe, which implies that all general physical laws should have the same form in any two coordinate systems that differ only by a rotation. It is therefore useful to classify physical quantities according to their transformation properties under coordinate rotations. Examples of scalars include the mass of an object, its electric charge, its volume, its surface area, the energy of a system, and its temperature. Other quantities have a direction and thus are vectors, such as the force exerted on a body, its velocity, its acceleration, its angular momentum, and the electric and magnetic fields. Since the sum of two scalars is a scalar, and the sum of two vectors is a vector, it is important in the formulation of physical laws not to mix quantities that have different transformation properties under coordinate rotations. The sum of a vector and a scalar has no simple transformation properties; a law that equated a vector to a scalar would have different forms in different coordinate systems and would thus not be acceptable.