# Dimensions

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## Dimensions (mechanics)

Length, mass, time, or combinations of these quantities serving as an indication of the nature of a physical quantity. Quantities with the same dimensions can be expressed in the same units. For example, although speed can be expressed in various units such as miles/hour, feet/second, and meters/second, all these speed units involve the ratio of a length unit to a time unit; hence, the dimensions of speed are the ratio of length *L* to time *T*, usually stated as *LT*^{-1}. The dimensions of all mechanical quantities can be expressed in terms of *L*, *T*, and mass *M*. The validity of algebraic equations involving physical quantities can be tested by a process called dimensional analysis; the terms on the two sides of any valid equation must have the same dimensions. *See* Dimensional analysis, Units of measurement

## Dimensions

The dimensions of a physical quantity show by what factor the unit of the physical quantity is changed when the units of the quantities adopted as fundamental in the given system are altered. The dimensions are expressed by a monomial consisting of the product of various powers of the fundamental units, which are represented by generalized symbols; the exponents may be positive or negative integers or fractions. Thus, the dimensions of velocity are *LT*^{-1}, where *T* represents the dimension of time and *L* the dimension of length. These symbols denote the units of time and length regardless of the specific size of the units. The units may be, for example, a second, minute, or hour and a meter or centimeter. In many cases, the dimensions permit the relations between corresponding quantities to be established.