College Physics

Figure 3.31To add vectorsAandB, first determine the horizontal and vertical components of each vector. These are the dotted vectorsAx,Ay,BxandByshown

in the image.

Step 2.Find the components of the resultant along each axis by adding the components of the individual vectors along that axis.That is, as shown in

Figure 3.32,

Rx=Ax+Bx (3.12)

and

Ry=Ay+By. (3.13)

Figure 3.32The magnitude of the vectorsAxandBxadd to give the magnitudeRxof the resultant vector in the horizontal direction. Similarly, the magnitudes of the

vectorsAyandByadd to give the magnitudeRyof the resultant vector in the vertical direction.

Components along the same axis, say thex-axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The

same is true for components along they-axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the

second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes

it easier to add them. Now that the components ofRare known, its magnitude and direction can be found.

Step 3.To get the magnitudeRof the resultant, use the Pythagorean theorem:

(3.14)

R= Rx^2 +R^2 y.

Step 4.To get the direction of the resultant:

θ= tan−1(R (3.15)

y/Rx).

The following example illustrates this technique for adding vectors using perpendicular components.

Example 3.3 Adding Vectors Using Analytical Methods

Add the vectorAto the vectorBshown inFigure 3.33, using perpendicular components along thex- andy-axes. Thex- andy-axes are along

the east–west and north–south directions, respectively. VectorArepresents the first leg of a walk in which a person walks53.0 min a

direction20.0ºnorth of east. VectorBrepresents the second leg, a displacement of34.0 min a direction63.0ºnorth of east.

98 CHAPTER 3 | TWO-DIMENSIONAL KINEMATICS

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