# Motion in Two Dimensions

## Vector Addition and Subtraction: Analytical Methods


### Section Key Terms


### Components of Vectors

For the analytical method of vector addition and subtraction, we use some simple geometry and trigonometry, instead of using a ruler and protractor as we did for graphical methods. However, the graphical method will still come in handy to visualize the problem by drawing vectors using the head-to-tail method. The analytical method is more accurate than the graphical method, which is limited by the precision of the drawing. For a refresher on the definitions of the sine, cosine, and tangent of an angle, see .
Since, by definition,  , we can find the length x if we know h and    by using   . Similarly, we can find the length of y by using   . These trigonometric relationships are useful for adding vectors.
When a vector acts in more than one dimension, it is useful to break it down into its x and y components. For a two-dimensional vector, a component is a piece of a vector that points in either the x- or y-direction. Every 2-d vector can be expressed as a sum of its x and y components.
For example, given a vector like 
       in , we may want to find what two perpendicular vectors, 
       and 
      , add to produce it. In this example, 
       and 
       form a right triangle, meaning that the angle between them is 90 degrees. This is a common situation in physics and happens to be the least complicated situation trigonometrically.
 
       and 
       are defined to be the components of 
       along the x- and y-axes. The three vectors, 
      , 
      , and 
      , form a right triangle.
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
where 
         is the magnitude of A in the x-direction, 
         is the magnitude of A in the y-direction, and 
         is the angle of the resultant with respect to the x-axis, as shown in .
Suppose, for example, that 
       is the vector representing the total displacement of the person walking in a city, as illustrated in .
Then A = 10.3 blocks and  
      , so that 
This magnitude indicates that the walker has traveled 9 blocks to the east—in other words, a 9-block eastward displacement. Similarly, 
indicating that the walker has traveled 5 blocks to the north—a 5-block northward displacement.

### Analytical Method of Vector Addition and Subtraction

Calculating a resultant vector (or vector addition) is the reverse of breaking the resultant down into its components. If the perpendicular components  
          and  
          of a vector  
          are known, then we can find  
          analytically. How do we do this? Since, by definition, 
we solve for  
          to find the direction of the resultant.
Note that  gives an angle in the first quadrant if 
       and in the fourth quadrant if 
      . If, in fact, both  and  are negative, or if  is negative and  positive, then , measured from the positive  direction, is 
      .
Since this is a right triangle, the Pythagorean theorem (x2 + y2 = h2) for finding the hypotenuse applies. In this case, it becomes
Solving for A gives
In summary, to find the magnitude  
          and direction  
          of a vector from its perpendicular components  
          and  
        , as illustrated in , we use the following relationships:
Sometimes, the vectors added are not perfectly perpendicular to one another. An example of this is the case below, where the vectors  
          and  
          are added to produce the resultant  
          as illustrated in .
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. The person could have walked straight ahead 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 
         . 
1. Draw in the x and y components of each vector (including the resultant) with a dashed line. Use the equations 
           and 
           to find the components. In , these components are 
          , 
          , 
          , and 
           Vector 
           makes an angle of 
           with the x-axis, and vector 
           makes and angle of 
           with its own x-axis (which is slightly above the x-axis used by vector A).
          
            
2. Find the x component of the resultant by adding the x component of the vectors
          and find the 
          
            
          Now that we know the components of  
3. To get the magnitude of the resultant R, use the Pythagorean theorem.
          
4. To get the direction of the resultant
          

### Using the Analytical Method of Vector Addition and Subtraction to Solve Problems

 uses the analytical method to add vectors.

### Practice Problems


### Section Summary

1. The analytical method of vector addition and subtraction uses 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:
      
        
      

### Key Equations


### Check Your Understanding


### Concept Items


### Problems


### Critical Thinking


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
