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.
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.
Fig. 3-2 (a) AC is the vector sum of the vectors AB and BC. (b) The same vectors relabeled.
Fig. 3-3 The two vectors and 
can be added in either order; see Eq. 3-2.
Fig. 3-4 The three vectors , and 
can be grouped in any way as they are added; see Eq. 3-3.
We can represent the relation among the three vectors in Fig. 3-2b with the vector equation
which says that the vector 
is the vector sum of vectors and . 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 and geometrically. (1) On paper, sketch vector to some convenient scale and at the proper angle. (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. (3) The vector sum is the vector that extends from the tail of to the head of .
Vector addition, defined in this way, has two important properties. First, the order of addition does not matter. Adding to gives the same result as adding to (Fig. 3-3); that is,
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 , , and , we can add and first and then add their vector sum to . We can also add and first and then add that sum to . We get the same result either way, as shown in Fig. 3-4. That is,
Fig. 3-5 The vectors and − have the same magnitude and opposite directions.
The vector − 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
Thus, adding − has the effect of subtracting . We use this property to define the difference between two vectors: let 
= − . Then
that is, we find the difference vector by adding the vector − to the vector . 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 , we can rearrange the equation as
Fig. 3-6 (a) Vectors , , and −. (b) To subtract vector from vector , add vector − to vector .
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.
CHECKPOINT 1 The magnitudes of displacements 
and 
are 3 m and 4 m, respectively, and = + . Considering various orientations of and , what is (a) the maximum possible magnitude for 
and (b) the minimum possible magnitude?
In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) , 2.0 km due east (directly toward the east); (b) , 2.0 km 30° north of east (at an angle of 30° toward the north from due east); (c) , 1.0 km due west. Alternatively, you may substitute either − for or − for . What is the greatest distance you can be from base camp at the end of the third displacement?
Solution: Using a convenient scale, we draw vectors , , , −, and − as in Fig. 3-7a. We then mentally slide the vectors over the page, connecting three of them at a time in head-to-tail arrangements to find their vector sum . The tail of the first vector represents base camp. The head of the third vector represents the point at which you stop. The vector sum extends from the tail of the first vector to the head of the third vector. Its magnitude d is your distance from base camp.
We find that distance d is greatest for a head-to-tail arrangement of vectors , , and −. They can be in any order, because their vector sum is the same for any order. The order shown in Fig. 3-7b is for the vector sum
Fig. 3-7 (a) Displacement vectors; three are to be used. (b) Your distance from base camp is greatest if you undergo displacements , , and −, in any order.
Using the scale given in Fig. 3-7a, we measure the length d of this vector sum, finding
d = 4.8 m. (Answer)
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