Vector Addition and Subtraction: Analytical Methods
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
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
If the vector
The magnitudes of the vector components
Suppose, for example, that
We can use the relationships
Calculating a Resultant Vector
If the perpendicular components
The magnitude and direction of the resultant vector can be determined once the horizontal and vertical components
Note that the equation
Adding Vectors Using Analytical Methods
To see how to add vectors using perpendicular components, consider Figure, in which the vectors
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
To add vectors
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,
The magnitude of the vectors
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
Step 3. To get the magnitude
Step 4. To get the direction of the resultant:
The following example illustrates this technique for adding vectors using perpendicular components.
Add the vector
The components of
Following the method outlined above, we first find the components of
Similarly, the y-components are found using
The x- and y-components of the resultant are thus
Now we can find the magnitude of the resultant by using the Pythagorean theorem:
Finally, we find the direction of the resultant:
Using analytical methods, we see that the magnitude of
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,
and the rest of the method outlined above is identical to that for addition. (See Figure.)
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.<figure class="ui-has-child-figcaption" id="import-auto-id1165298841604" style="width: 840px;">
The subtraction of the two vectors shown in Figure. The components of
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.<figure class="ui-has-child-figcaption" id="eip-id3192946" style="width: 660px;">
- 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
Aand Busing 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
Step 2: Add the horizontal and vertical components of each vector to determine the components
Rxand Ryof the resultant vector, R: Rx=Ax+Bx
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).
Suppose you add two vectors
Give an example of a nonzero vector that has a component of zero.
Explain why a vector cannot have a component greater than its own magnitude.
If the vectors
Problems & Exercises
Find the following for path C in Figure: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.<figure class="ui-has-child-figcaption" id="import-auto-id1165298863773" style="width: 782px;">
The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.</figcaption> </figure>
(a) 1.56 km
(b) 120 m east
Find the following for path D in Figure: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.
Find the north and east components of the displacement from San Francisco to Sacramento shown in Figure.<figure id="import-auto-id1165298797444" style="width: 782px;">
North-component 87.0 km, east-component 87.0 km
Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements
The two displacements
Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.
Repeat Exercise using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result—that is,
30.8 m, 35.8 west of north
Do Exercise again using analytical techniques and change the second leg of the walk to
A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors
18.4 km south, then 26.2 km west(b) 31.5 km at
A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as
<figure id="import-auto-id1165298543237" style="width: 782px;">
In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines:
Suppose a pilot flies
- analytical method
- the method of determining the magnitude and direction of a resultant vector using the Pythagorean theorem and trigonometric identities
Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).