To the mathematician, a vector is an abstract “object,” in the same way that a number or a group or a knot or a tensor is an object. A vector is defined by how many numbers it takes to specify it, how it combines with other vectors or with numbers, and how it transforms from one reference frame to another. A vector is not itself a physical “thing.” It is an abstraction that can be manipulated in accordance with its defining rules. If the physicist discovers a physical thing whose behavior in the real world exactly matches the behavior of a mathematical vector, the physicist calls it a vector quantity. There are numerous such quantities: force, velocity, momentum, torque, angular momentum, and so on.
So if a quantity that can be measured and that enters into our equations of physics “behaves” in every way like a mathematical vector, we call in a vector quantity. The teacher who calls force (or any other vector quantity) a vector can be forgiven. That teacher should be aware of the difference but can choose the shorthand name. The student, too, should be made aware of the difference, but can practice what our colleagues in the English department call synecdoche.
In a similar way, a scalar quantity is a physical quantity that “behaves” like a mathematical scalar, or, equivalently, like a real number. Mass and electric charge are examples of scalar quantities. A mass of 2 kg added to a mass of 3 kg produces a mass of 5 kg, in exactly the same way that the arithmetic sum of 2 and 3 is 5. But a force of 2 N combined with a force of 3 N does not necessarily produce a force of 5 N. So force is not a scalar quantity.
In addition to these two ideas that need to be kept straight—the mathematical object and the physical quantity—there’s a third idea that deserves attention: the geometrical representation. With numbers, the geometrical representation is interesting but hardly necessary—a number line. With vectors, the geometrical representation is almost a necessity. Since vectors are somewhat more abstract than numbers, they defy easy visualization. Only with the help of arrow diagrams can we think easily about vectors and vector arithmetic. A vector quantity, as defined above, is a physical concept that behaves like a vector, in an exact mathematical sense. Sometimes a vector quantity is defined roughly as a quantity with both magnitude and direction. This is not quite an adequate definition, for to be a vector quantity it must not only have magnitude and direction. It must also follow precisely the laws of vector arithmetic when it enters into a physical law.
Velocity, for instance, is a quantity with both magnitude and direction. So long as its magnitude is small compared with the speed of light, it behaves like a vector. But at enormous speeds its behavior deviates from that of a vector. Although it has magnitude and direction, it is no longer a vector quantity. This is an exceptional case, however. The student who assumes that any quantity with magnitude and direction is a vector quantity will almost always be correct. Even the exception, velocity, is most often treated as a vector quantity, for that is what it is in classical physics. The fact that a quantity once believed to be a vector quantity turns out not to be a vector quantity emphasizes a very important point about all physical concepts. The association of any physical quantity with a certain mathematical quantity— be it number, vector, or something else—is based purely on experiment and is always subject to later change.