A large number of physical quantities used in physics need to have a direction specified as well as a size in order to completely describe them.  Such quantities are called vectors.  Vectors which will be studied in an introductory physics course include: displacement, velocity, acceleration, force, torque, momentum, angular momentum, gravitational field, electric field, magnetic field, and current.  Vectors can be added, subtracted and multiplied.  In this section the rules governing the mathematical operations with vectors will be summarized.  A quantity which has a magnitude but no direction is called a scalar.  Scalars which will be studied in physics include: mass, work, energy, temperature, time, voltage, density, and volume.  Weight, which is the force of gravity, is a vector whose direction is always toward the center of the earth.  Following are definitions of some of the important terms related to operations with vectors.

A vector is specified by giving its magnitude and a direction.  Sometimes the direction is given as a geographical direction (north, south, etc).  It can also be given by specifying the angle the vector makes with a specified reference line such as an axis of a cartesian coordinate system or with the vertical or horizontal.  Remember if the answer to a problem is a vector then you must specify both the magnitude and direction for the answer to be correct.

A vector may be represented by an arrow drawn in the proper direction with a length proportional to the magnitude of the vector.  While we could work with scale diagrams we rarely do, choosing instead to sketch vectors in approximately the proper direction and of the proper length and then using mathematical methods to find any unknown vectors.  The symbol for a vector is usually a letter with a short arrow drawn over it.  However, it is also common to write vectors in bold text. This will be the convention used in the text of this book. A is an exmple of a vector.  When only the magnitude of a vector is desired, the same letter will be used without the vector symbol or in normal print.

Two vectors are equal if and only if they have the same magnitude and the same direction.  Note that their locations on a page or in space are immaterial as long as they are parallel and the same length.

A vector can be added to another vector, multiplied by a scalar, or multiplied by another vector.  There are two types of vector multiplication; one yields another vector and the other yields a scalar.  Subtraction of vectors is defined as the addition of a vector and the negative of a second vector.  All of these operations will be discussed in following sections.

A vector can be broken up into components.  Any two (or more) vectors which add up to a given vector are said to be components of that vector.  A vector has many sets of components, just as there are many numbers which add up to a given number.  For example 8 and 2; 7 and 3; and 2, 3 and 5 are all components of the number 10.  The most useful set of components of a vector is a set of two components which are perpendicular to each other.  They will normally be parallel to the coordinate axis used.

The sum of two or more vectors is often called the resultant.