3.3: Vector Addition and Subtraction- Analytical Methods
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Learning Objectives
By the end of this section, you will be able to:
- Understand the rules of vector addition and subtraction using analytical methods.
- Apply analytical methods to determine vertical and horizontal component vectors.
- Apply analytical methods to determine the magnitude and direction of a resultant vector.
Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.
Resolving a Vector into Perpendicular Components
Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like A in Figure 3.3.1, we may wish to find which two perpendicular vectors, Ax and Ay, add to produce it.

Figure 3.3.1:The vector A, with its tail at the origin of an x, y-coordinate system, is shown together with its x- and y-components, Ax and Ay. These vectors form a right triangle. The analytical relationships among these vectors are summarized below.
Ax and Ay are defined to be the components of A along the x- and y-axes. The three vectors A,Ax, and Ay form a right triangle:
Ax+Ay=A.
Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if Ax=3m east, Ay=4m north, and A=5mnorth-east, then it is true that the vectors Ax+Ay=A. However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,
3m+4m≠5m
Thus,
Ax+Ay≠A
If the vector A is known, then its magnitude A (its length) and its angle θ (its direction) are known. To find Ax and Ay, its x- and y-components, we use the following relationships for a right triangle.
Ax=Acosθ
and
Ay=Asinθ.

Suppose, for example, that A is the vector representing the total displacement of the person walking in a city considered in Kinematics in Two Dimensions: An Introduction and Vector Addition and Subtraction: Graphical Methods.

Figure 3.3.3: We can use the relationships Ax=Acosθ and Ay=Asinθ to determine the magnitude of the horizontal and vertical component vectors in this example.
Then A=10.3 blocks and θ=29.1º, so that
Ax=Acosθ=(10.3blocks)(cos29.1º)=9.0blocks
Ay=Asinθ=(10.3blocks)(sin29.1º)=5.0blocks
Calculating a Resultant Vector
If the perpendicular components Ax and Ay of a vector A are known, then A can also be found analytically. To find the magnitude A and direction θ of a vector from its perpendicular components Ax and Ay, we use the following relationships:
A=√Ax2+Ay2
θ=tan−1(Ay/Ax)

Figure 3.3.4: The magnitude and direction of the resultant vector can be determined once the horizontal and vertical components Ax and Ay have been determined.
Note that the equation A=√A2x+A2y is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if Ax and Ay are 9 and 5 blocks, respectively, then A=√92+52=10.3 blocks, again consistent with the example of the person walking in a city. Finally, the direction is θ=tan–1(5/9)=29.1º, as before.
DETERMINING VECTORS AND VECTOR COMPONENTS WITH ANALYTICAL METHODS
Equations Ax=Acosθ and Ay=Asinθ are used to find the perpendicular components of a vector—that is, to go from A and θ to Ax and Ay. Equations A=√A2x+A2y and θ=tan–1(Ay/Ax) are used to find a vector from its perpendicular components—that is, to go from Ax and Ay to A and θ. Both processes are crucial to analytical methods of vector addition and subtraction.
Adding Vectors Using Analytical Methods
To see how to add vectors using perpendicular components, consider Figure 3.3.5, in which the vectors A and B are added to produce the resultant R.

Figure 3.3.5: Vectors A andB are two legs of a walk, and R is the resultant or total displacement. You can use analytical methods to determine the magnitude and direction of R.
If A and B represent two legs of a walk (two displacements), then R is the total displacement. The person taking the walk ends up at the tip of R. 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, Rx and Ry. If we know Rx and Ry, we can find R and θ using the equations A=√A2x+A2y and θ=tan–1(Ay/Ax). 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 Ax=Acosθ and Ay=Asinθ to find the components. In Figure, these components are Ax,Ay,Bx, and By. The angles that vectors A and B make with the x-axis are θA and θB, respectively.

Figure 3.3.6: To add vectors A and B, first determine the horizontal and vertical components of each vector. These are the dotted vectors Ax,Ay,Bx and B−y shown 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 \(\PageIndex{7}\,
Rx=Ax+Bx
and
Ry=Ay+By.

Figure 3.3.7: The magnitude of the vectors Ax and Bx add to give the magnitude Rx of the resultant vector in the horizontal direction. Similarly, the magnitudes of the vectors Ay and By add to give the magnitude Ry of the resultant vector in the vertical direction.
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 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 of R are known, its magnitude and direction can be found.
Step 3. To get the magnitude R of the resultant, use the Pythagorean theorem:
R=√R2x+R2y
Step 4. To get the direction of the resultant:
θ=tan−1(Ry/Rx)
The following example illustrates this technique for adding vectors using perpendicular components.
Example 3.3.1: Adding Vectors Using Analytical Methods
Add the vector A to the vector B shown in Figure, 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 A represents the first leg of a walk in which a person walks 53.0m in a direction 20.0º north of east. Vector B represents the second leg, a displacement of 34.0m in a direction 63.0º north of east.
Figure 3.3.8: Vector A has magnitude 53.0m and direction 20.0º north of the x-axis. Vector B has magnitude 34.0m and direction 63.0º north of the x-axis. You can use analytical methods to determine the magnitude and direction of R.
Strategy
The components of A and B 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 A and B along the x- and y-axes. Note that A=53.0m,θA=20.0º,B=34.0m, and θB=63.0º. We find the x-components by using Ax=Acosθ, which gives
Ax=AcosθA=(53.0m)(cos20.0º)(53.0m)(0.940)=49.8m
and
Bx=BcosθB=(34.0m)(cos63.0º)(34.0m)(0.454)=15.4m.
Similarly, the y-components are found using Ay=AsinθA:
Ay=AsinθA=(53.0m)(sin20.0º)(53.0m)(0.342)=18.1m
and
By=BsinθB=(34.0m)(sin63.0º)(34.0m)(0.891)=30.3m.
The x- and y-components of the resultant are thus
Rx=Ax+Bx=49.8m+15.4m=65.2m
and
Ry=Ay+By=18.1m+30.3m=48.4m.
Now we can find the magnitude of the resultant by using the Pythagorean theorem:
R=√R2x+R2y=√(65.2)2+(48.4)2m
so that
R=81.2m.
Finally, we find the direction of the resultant:
θ=tan−1(Ry/Rx)=+tan−1(48.4/65.2).
Thus,
θ=tan−1(0.742)=36.6º.

Figure 3.3.9: Using analytical methods, we see that the magnitude of R is 81.2m and its direction is 36.6º north of east.
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, A−B≡A+(–B). Thus, the method for the subtraction of vectors using perpendicular components is identical to that for addition. The components of –B are the negatives of the components of B. The x- and y-components of the resultant A−B=R are thus
Rx=Ax+(–Bx)
and
Ry=Ay+(–By)
and the rest of the method outlined above is identical to that for addition. (See Figure 3.3.10.)
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.

PHET EXPLORATIONS: VECTOR ADDITION
Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.
- The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.
- The steps to add vectors A and B using the analytical method are as follows:
Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equations
Ax=Acosθ
Bx=Bcosθ
and
Ay=Asinθ
By=Bsinθ.
Step 2: Add the horizontal and vertical components of each vector to determine the components Rx and Ry of the resultant vector, R:
Rx=Ax+Bx
and
Ry=Ay+By.
Step 3: Use the Pythagorean theorem to determine the magnitude, R, of the resultant vector R:
R=√R2x+R2y.
Step 4: Use a trigonometric identity to determine the direction, θ, of R:
θ=tan−1(Ry/Rx).
Glossary
- analytical method
- the method of determining the magnitude and direction of a resultant vector using the Pythagorean theorem and trigonometric identities