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When we speak of the dimensionsof a quantity, we are referring to the type of
base units or base quantities that make it up. The dimensions of area, for
example, are always length squared, abbreviated using square brackets;
the units can be square meters, square feet, and so on. Velocity, on the
other hand, can be measured in units of or but the dimensions are always a length [L] divided by a time [T]: that is,
The formula for a quantity may be different in different cases, but the dimensions remain the same. For example, the area of a triangle of base band height h
is whereas the area of a circle of radius ris The formulas
are different in the two cases, but the dimensions of area are always
Dimensions can be used as a help in working out relationships, a procedure
referred to as dimensional analysis. One useful technique is the use of dimensions to check if a relationship is incorrect. Note that we add or subtract
quantities only if they have the same dimensions (we don’t add centimeters
and hours); and the quantities on each side of an equals sign must have the
same dimensions. (In numerical calculations, the units must also be the same on
both sides of an equation.)