Adding Vectors Using Analytical Methods
To see how to add vectors using perpendicular components, consider Figure 3.30, in which the vectors and are added to produce the resultant .
If and represent two legs of a walk (two displacements), then is the total displacement. The person taking the walk ends up at the tip of There are many ways to arrive at the same point. In particular, the person could have walked first in the x-direction and then in the y-direction. Those paths are the x- and y-components of the resultant, and . If we know
and , we can find
and
using the equations
and
. When you use the analytical method of vector addition, you can determine the components or the magnitude and direction of a vector.
Step 1. Identify the x- and y-axes that will be used in the problem. Then, find the components of each vector to be added along the chosen perpendicular axes. Use the equations
and
to find the components. In Figure 3.31, these components are
,
,
, and
. The angles that vectors and make with the x-axis are and , respectively.
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,
3.12
and
3.13
Components along the same axis, say the x-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 the y-axis. For example, a 9-block eastward walk could be taken in two legs, the first three blocks east and the second six blocks east, for a total of nine, 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 of are known, its magnitude and direction can be found.
Step 3. To get the magnitude of the resultant, use the Pythagorean theorem.
3.14 Step 4. To get the direction of the resultant
3.15 The following example illustrates this technique for adding vectors using perpendicular components.
Example 3.3 Adding Vectors Using Analytical Methods
Add the vector to the vector shown in Figure 3.33, using perpendicular components along the x- and y-axes. The x- and y-axes are along the east-west and north-south directions, respectively. Vector represents the first leg of a walk in which a person walks in a direction north of east. Vector represents the second leg, a displacement of in a direction north of east.
Strategy
The components of and along the x- and y-axes represent walking due east and due north to get to the same ending point. Once found, they are combined to produce the resultant.
Solution
Following the method outlined above, we first find the components of
and
along the x- and y-axes. Note that
,
,
, and .
We find the x-components by using , which gives
3.16
and
3.17
Similarly, the y-components are found using .
3.18
and
3.19
The x- and y-components of the resultant are thus
3.20
and
3.21 Now we can find the magnitude of the resultant by using the Pythagorean theorem
3.22
so that
3.23 Finally, we find the direction of the resultant
3.24 Thus,
3.25
Discussion
This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar—it is just the addition of a negative vector.
Subtraction of vectors is accomplished by the addition of a negative vector. That is, . Thus, the method for the subtraction of vectors using perpendicular components is identical to that for addition. The components of
are the negatives of the components of
. The x- and y-components of the resultant are thus
3.26
and
3.27
and the rest of the method outlined above is identical to that for addition (see Figure 3.35).
Analyzing vectors using perpendicular components is very useful in many areas of physics, because perpendicular quantities are often independent of one another. The next module, Projectile Motion, is one of many in which using perpendicular components helps make the picture clear and simplifies the physics.