# Two-Dimensional Kinematics

## Vector Addition and Subtraction: Analytical Methods


### Learning Objectives

By the end of this section, you will be able to:
1. Understand the rules of vector addition and subtraction using analytical methods.
2. Apply analytical methods to determine vertical and horizontal component vectors.
3. 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  in , we may wish to find which two perpendicular vectors,  and , add to produce it.
 and   are defined to be the components of  along the x- and y-axes. The three vectors , , and  form a right triangle:
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  east,

 north, and

 north-east, then it is true that the vectors . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,
Thus,
If the vector  is known, then its magnitude  (its length) and its angle  
        
           (its direction) are known. To find  and , its x- and y-components, we use the following relationships for a right triangle.
and
Suppose, for example, that  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.
Then  blocks and

 
  
, so that



### Calculating a Resultant Vector

If the perpendicular components  and  of a vector  are known, then  can also be found analytically. To find the magnitude  and direction  
        
           of a vector from its perpendicular components  and , relative to the x-axis, we use the following relationships:
Note that the equation

 
   is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if  and  are 9 and 5 blocks, respectively, then  blocks, again consistent with the example of the person walking in a city. Finally, the direction is

 
  

, as before.



### Adding Vectors Using Analytical Methods

To see how to add vectors using perpendicular components, consider , 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.
 Use the equations
 and

 to find the components. In , these components are

,

,

, and

. The angles that vectors  and  make with the x-axis are  and , respectively.
 That is, as shown in ,
and
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  are known, its magnitude and direction can be found.


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

### Summary

1. 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.
2. The steps to add vectors  and  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
    and
    Step 2: Add the horizontal and vertical components of each vector to determine the components 
      
    and
    
    Step 3: Use the Pythagorean theorem to determine the magnitude, 
    Step 4: Use a trigonometric identity to determine the direction, 
    

### Conceptual Questions


### Problems & Exercises
