1.5. Scalars and Vectors

The volume of water in a swimming pool might be 50 cubic meters, or the winning time of a race could be 11.3 seconds. In cases like these, only the size of the numbers matters. In other words, how much volume or time is there? The “50” specifies the amount of water in units of cubic meters, while the “11.3” specifies the amount of time in seconds. Volume and time are examples of scalar quantities. A scalar quantity is one that can be described with a single number (including any units) giving its size or magnitude. Some other common scalars are temperature (e.g., 20°C) and mass (e.g., 85 kg).

While many quantities in physics are scalars, there are also many that are not, and for these quantities the magnitude tells only part of the story. Consider Figure 1.8, which depicts a car that has moved 2 km along a straight line from start to finish. When describing the motion, it is incomplete to say that “the car moved a distance of 2 km.” This statement would indicate only that the car ends up somewhere on a circle whose center is at the starting point and whose radius is 2 km. A complete description must include the direction along with the distance, as in the statement “the car moved a distance of 2 km in a direction 30° north of east.” A quantity that deals inherently with both magnitude and direction is called a vector quantity. Because direction is an important characteristic of vectors, arrows are used to represent them; the direction of the arrow gives the direction of the vector. The colored arrow in Figure 1.8, for example, is called the displacement vector, because it shows how the car is displaced from its starting point. Chapter 2 discusses this particular vector.

Figure 1.8 A vector quantity has a magnitude and a direction. The arrow in this drawing represents a displacement vector.

The length of the arrow in Figure 1.8 represents the magnitude of the displacement vector. If the car had moved 4 km instead of 2 km from the starting point, the arrow would have been drawn twice as long. By convention, the length of a vector arrow is proportional to the magnitude of the vector.

In physics there are many important kinds of vectors, and the practice of using the length of an arrow to represent the magnitude of a vector applies to each of them. All forces, for instance, are vectors. In common usage a force is a push or a pull, and the direction in which a force acts is just as important as the strength or magnitude of the force. The magnitude of a force is measured in SI units called newtons (N). An arrow representing a force of 20 newtons is drawn twice as long as one representing a force of 10 newtons.

The fundamental distinction between scalars and vectors is the characteristic of direction. Vectors have it, and scalars do not. Conceptual Example 5 helps to clarify this distinction and explains what is meant by the “direction” of a vector.

The velocity of this in-line skater is another example of a vector quantity, because the velocity has a magnitude (the speed of the skater) and a direction. (© Scott Markewitz/Taxi/Getty Images)

Conceptual Example 5 Vectors, Scalars, and the Role of Plus and Minus Signs

There are places where the temperature is +20°C at one time of the year and –20°C at another time. Do the plus and minus signs that signify positive and negative temperatures imply that temperature is a vector quantity?

Reasoning and Solution A vector has a physical direction associated with it, due east or due west, for example. The question, then, is whether such a direction is associated with temperature. In particular, do the plus and minus signs that go along with temperature imply this kind of direction? On a thermometer, the algebraic signs simply mean that the temperature is a number less than or greater than zero on the scale and have nothing to do with east, west, or any other physical direction. Temperature, then, is not a vector. It is a scalar, and scalars can sometimes be negative. The fact that a quantity is positive or negative does not necessarily mean that the quantity is a scalar or a vector.

Often, for the sake of convenience, quantities such as volume, time, displacement, and force are represented by symbols. This text follows the usual practice of writing vectors in boldface symbols (this is boldface) and writing scalars in italic symbols (this is italic). Thus, a displacement vector is written as “ A = 750 m, due east,” where the A is a boldface symbol. By itself, however, separated from the direction, the magnitude of this vector is a scalar quantity. Therefore, the magnitude is written as “ A = 750 m,” where the A is an italic symbol.

Check Your Understanding 1

Which of the following statements, if any, involves a vector? (a) I walked 2 miles along the beach. (b) I walked 2 miles due north along the beach. (c) I jumped off a cliff and hit the water traveling at 17 miles per hour. (d) I jumped off a cliff and hit the water traveling straight down at 17 miles per hour. (e) My bank account shows a negative balance of –25 dollars.

These questions deal with the concepts of vectors and scalars, and the difference between them.

For similar questions (including calculational counterparts), consult Self-Assessment Test 1.1. This test is described at the end of Section 1.6