Vectors and Scalars by David Halliday , Robert Resnick

A particle moving along a straight line can move in only two directions. We can take its motion to be positive in one of these directions and negative in the other. For a particle moving in three dimensions, however, a plus sign or minus sign is no longer enough to indicate a direction. Instead, we must use a vector.

A vector has magnitude as well as direction, and vectors follow certain (vector) rules of combination, which we examine in this chapter. A vector quantity is a quantity that has both a magnitude and a direction and thus can be represented with a vector. Some physical quantities that are vector quantities are displacement, velocity, and acceleration.

Figure 3-1 (a) All three arrows have the same magnitude and direction and thus represent the same displacement.

Not all physical quantities involve a direction.Temperature, pressure, energy, mass, and time, for example, do not "point" in the spatial sense. We call such quantities scalars, and we deal with them by the rules of ordinary algebra. A single value, with a sign (as in a temperature of -40oF), specifies a scalar.

The simplest vector quantity is displacement, or change of position. A vector that represents a displacement is called, reasonably, a displacement vector. (Similarly, we have velocity vectors and acceleration vectors.) If a particle changes its position by moving from A to B in Fig. 3-1a, we say that it undergoes a displacement from A to B, which we represent with an arrow pointing from A to B. The arrow specifies the vector graphically. To distinguish vector symbols from other kinds of arrows in this book, we use the outline of a triangle as the arrowhead.

Figure 3-1 (b) All three paths connecting the two points correspond to the same displacement vector.

In Fig. 3-1a, the arrows from A to B, from A' to B', and from A" to B" have the same magnitude and direction. Thus, they specify identical displacement vectors and represent the same change of position for the particle. A vector can be shifted without changing its value if its length and direction are not changed.

The displacement vector tells us nothing about the actual path that the particle takes. In Fig. 3-1b, for example, all three paths connecting points A and B correspond to the same displacement vector, that of Fig. 3-1a. Displacement vectors represent only the overall effect of the motion, not the motion itself.

Figure 3-2 (a) AC is the vector sum of the vectors AB and BC.

Suppose that, as in the vector diagram of Fig. 3-2a, a particle moves from A to B and then later from B to C. We can represent its overall displacement (no matter what its actual path) with two successive displacement vectors, AB and BC. The net displacement of these two displacements is a single displacement from A to C. We call AC the vector sum (or resultant) of the vectors AB and BC. This sum is not the usual algebraic sum.

Figure 3-2 (b) The same vectors relabeled.

In Fig. 3-2b, we redraw the vectors of Fig. 3-2a and relabel them in the way that we shall use from now on, namely, with an arrow over an italic symbol, as in. If we want to indicate only the magnitude of the vector (a quantity that lacks a sign or direction), we shall use the italic symbol, as in a, b, and s. (You can use just a handwritten symbol.) A symbol with an overhead arrow always implies both properties of a vector, magnitude and direction. We can represent the relation among the three vectors in Fig. 3-2b with the vector equation

$\stackrel{\to }{s}=\stackrel{\to }{a}+\stackrel{\to }{b}$ (3-1)

which says that the vector $\stackrel{\to }{s}$ is the vector sum of vectors . The symbol + in Eq. 3-1 and the words "sum" and "add" have different meanings for vectors than they do in the usual algebra because they involve both magnitude and direction.

Figure 3-2 suggests a procedure for adding two-dimensional vectors geometrically. (1) On paper, sketch vector $\stackrel{\to }{a}$ to some convenient scale and at the proper angle. (2) Sketch vector $\stackrel{\to }{b}$ to the same scale, with its tail at the head of vector $\stackrel{\to }{a}$, again at the proper angle. (3) The vector sum $\stackrel{\to }{s}$ is the vector that extends from the tail of $\stackrel{\to }{a}$ to the head of $\stackrel{\to }{b}$.

The two vectors and can be added in either order; see Eq. 3-2.

Properties. Vector addition, defined in this way, has two important properties. First, the order of addition does not matter. Adding gives the same result as adding (Fig. 3-3); that is,

$\stackrel{\to }{a}+\stackrel{\to }{b}=\stackrel{\to }{b}+\stackrel{\to }{a}$ (commutative law) (3-2)

Second, when there are more than two vectors, we can group them in any order as we add them. Thus, if we want to add vectors , we can add first and then add their vector sum to $\stackrel{\to }{c}$. We can also add first and then add that sum to $\stackrel{\to }{a}$. We get the same result either way, as shown in Fig. 3-4. That is,

The three vectors can be grouped in any way as they are added; see Eq. 3-3.

$\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)+\stackrel{\to }{c}=\stackrel{\to }{a}+\left(\stackrel{\to }{b}\right)+\stackrel{\to }{c}$ (associative law) (3-3).

figure 3-4 The three vectors a, b, c and can be grouped in any way as they are added;see Eq 3-3

Figure 3-5 The vectors have the same magnitude and opposite directions

The vector $\stackrel{\to }{\mathrm{-b}}$ is a vector with the same magnitude as but the opposite direction (see Fig. 3-5). Adding the two vectors in Fig. 3-5 would yield

$\stackrel{\to }{b}+\left(\stackrel{\to }{\mathrm{-b}}\right)=0$

Thus, adding $\stackrel{\to }{\mathrm{-b}}$ has the effect of subtracting $\stackrel{\to }{b}$. We use this property to define the difference between two vectors: let

$\stackrel{\to }{d}=\stackrel{\to }{a}-\stackrel{\to }{b}$. Then

$\stackrel{\to }{d}=\stackrel{\to }{a}-\stackrel{\to }{b}=\stackrel{\to }{a}+\left(\stackrel{\to }{\mathrm{-b}}\right)$ (vector subtraction) (3-4)

Figure 3-6 (a)

that is, we find the difference vector $\stackrel{\to }{d}$ by adding the vector $\stackrel{\to }{\mathrm{-b}}$ to the vector $\stackrel{\to }{a}$. Figure 3-6 shows how this is done geometrically.

As in the usual algebra, we can move a term that includes a vector symbol from one side of a vector equation to the other, but we must change its sign. For example, if we are given Eq. 3-4 and need to solve for $\stackrel{\to }{a}$, we can rearrange the equation as

$\stackrel{\to }{d}+\stackrel{\to }{b}=\stackrel{\to }{a}$

or

$\stackrel{\to }{a}=\stackrel{\to }{d}+\stackrel{\to }{b}$

Figure 3-6 (b) To subtract vector from vector, add vector $\stackrel{\to }{\mathrm{-b}}$ to vector $\stackrel{\to }{a}.$

Remember that, although we have used displacement vectors here, the rules for addition and subtraction hold for vectors of all kinds, whether they represent velocities, accelerations, or any other vector quantity. However, we can add only vectors of the same kind. For example, we can add two displacements, or two velocities, but adding a displacement and a velocity makes no sense. In the arithmetic of scalars, that would be like trying to add 21 s and 12 m.