# Scalar

A scalar is a number or measure, usually representing a physical quantity, that is not dependent upon direction. For example, distance is a scalar quantity since it may be expressed completely as a pure number without reference to spacial coordinates. Other examples of scalar quantities include mass, temperature, and time.

The term scalar originally referred to any quantity which is measurable on a scale. Take, for example, the numbers on a thermometer scale which measure temperature. These values require a positive or negative sign to indicate whether they are greater or less than zero, but they do not require an indication of direction because they have no component which describes their location in space. Such physical quantities which can be described completely by a pure number and which do not require a directional component are referred to as scalar quantities, or scalars. On the other hand, there are other physical measurements which have not only a magnitude (scalar) component but a directional component as well. For example, although we do not normally think of it as such, velocity is described not only by speed, but by the direction of movement too. Similarly, other physical quantities such as force, spin, and magnetism also involve spacial orientation. The mathematical expression used to describe such a combination of magnitude and direction is vector from the Latin word for "carrier." In its simplest form a vector can be described as a directed line segment. For example, if A and B are two distinct points, and AB is the line segment runs from A to B, then AB can also be called vector, v. Scalars are components of vectors which describe its magnitude, they provide information about the size of vectors. For example, for a vector representing velocity, the scalar which describes the magnitude of the movement is called speed. The direction of movement is described by an angle, usually designated as θ (theta).

The ability to separate scalar components from their corresponding vectors is important because it allows mathematical manipulation of the vectors. Two common mathematical manipulations involving scalars and vectors are scalar multiplication and vector multiplication. Scalar multiplication is achieved by multiplying a scalar and a vector together to give another vector with different magnitude. This is similar to multiplying a number by a scale factor to increase or decrease its value in proportion to its original value. In the example above, if the velocity is described by vector v and if c is a positive number, then cv is a different vector whose direction is that of v and whose length is c|v|. It should be noted that a negative value for c will result in a vector with the opposite direction of v. When a vector is multiplied by a scalar it can be made larger or smaller, or its direction can be reversed, but the angle of its direction relative to another vector will not change. Scalar multiplication is also employed in matrix algebra, where vectors are expressed in rectangular arrays known as matrices.

While scalar multiplication results in another vector, vector multiplication (in which two vectors are multiplied together) results in a scalar product. For example, if u and v are two different vectors with an angle between them of q, then multiplying the two gives the following: u . v = |uv|cosq. In this operation the value of the cos θ cancels out and the result is simply the scalar value, uv. The scalar product is sometimes called the dot product since a dot is used to symbolize the operation.

## Resources

### Books

Dunham, William. Journey Through Genius. New York: John Wiley, 1990.

Lloyd, G.E.R. Early Greek Science: Thales to Aristotle. New York: W.W. Norton, 1970.

Randy Schueller

## KEY TERMS

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Dot product

—Another term for scalar multiplication.

Scalar multiplication

—The multiplication of a scalar and a vector which yields another vector of different magnitude.

Scalar product

—The result of multiplication of two vectors, the value must be a real or complex number.

Vector multiplication

—The multiplication of two vectors to yield a scalar product.