# 29 Components of Vectors

OpenStax and Heath Hatch

Summary

This material is presented both as videos by Heath Hatch and with the relevant OpenStax textbook. Either method is fine. Regardless, by the end of this section, you should be able to:

- Apply analytical methods to determine vertical and horizontal component vectors.

**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 Figure 1, we may wish to find which two perpendicular vectors, and , add to produce it.

and are defined to be the components of 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 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

**Figure 2.**The magnitudes of the vector components

**A**and

_{x}**A**can be related to the resultant vector

_{y}**A**and the angle

**θ**with trigonometric identities. Here we see that

*and*

**A**_{x}=A cos θ**.**

*A*_{y}=A sinθSuppose, for example, that is the vector representing the total displacement of the person walking in a city considered in Chapter 3.1 Kinematics in Two Dimensions: An Introduction and Chapter 3.2 Vector Addition and Subtraction: Graphical Methods.

**Figure 3.**We can use the relationships

*and*

**A**_{x}=A cos θ**to determine the magnitude of the horizontal and vertical component vectors in this example.**

*A*_{y}=A sinθThen 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 we use the following relationships:

**Figure 4.**The magnitude and direction of the resultant vector can be determined once the horizontal and vertical components

*and*

**A**_{x}*have been determined.*

**A**_{y}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.

### DETERMINING VECTORS AND VECTOR COMPONENTS WITH ANALYTICAL METHODS

Equations and are used to find the perpendicular components of a vector—that is, to go from and to and Equations and are used to find a vector from its perpendicular components—that is, to go from and to and Both processes are crucial to analytical methods of vector addition and subtraction.

### Conceptual Questions

**1: **Give an example of a nonzero vector that has a component of zero.

**2: **Explain why a vector cannot have a component greater than its own magnitude.

### Problems & Exercises

**1: **Find the north and east components of the displacement from San Francisco to Sacramento shown in Figure 13.

**Figure 13.**

**2: **You drive in a straight line in a direction east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order.

**3: **You fly in a straight line in still air in the direction south of west. (a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.) (b) Find the distances you would have to fly first in a direction south of west and then in a direction west of north. These are the components of the displacement along a different set of axes—one rotated

## Glossary

- analytical method
- the method of determining the magnitude and direction of a resultant vector using the Pythagorean theorem and trigonometric identities

### Solutions

**Problems & Exercises
**

**1:**

North-component 87.0 km, east-component 87.0 km

**2:**

30.8 m, 35.8 west of north

**3:**

(a) 18.4 km south, then 26.2 km west

(b) 31.5 km at south of west, then 5.56 km at west of north