By the end of this section, you will be able to do the following:
- Define components of vectors
- Describe the analytical method of vector addition and subtraction
- Use the analytical method of vector addition and subtraction to solve problems
The learning objectives in this section will help your students master the following standards:
(3) Scientific processes. The student uses critical thinking, scientific reasoning, and problem solving to make informed decisions within and outside the classroom. The student is expected to:
- (F) express and interpret relationships symbolically in accordance with accepted theories to make predictions and solve problems mathematically, including problems requiring proportional reasoning and graphical vector addition
(4) Science concepts. The student knows and applies the laws governing motion in two dimensions for a variety of situations. The student is expected to:
- (E) develop and interpret free-body force diagrams;
- (F) identify and describe motion relative to different frames of reference.
In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Motion in Two Dimensions, as well as the following standards:
- (3) Scientific processes. The student uses critical thinking, scientific reasoning, and problem solving to make informed decisions within and outside the classroom. The student is expected to:
- (F) express and interpret relationships symbolically in accordance with accepted theories to make predictions and solve problems mathematically, including problems requiring proportional reasoning and graphical vector addition.
Section Key Terms
|analytical method||component (of a two-dimensional vector)|
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 Figure 5.17.
[BL][OL] Review trigonometric concepts of sine, cosine, tangent and the Pythagorean theorem.
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 Figure 5.18, 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:
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 Figure 5.19.
[BL][OL][AL] Derive the formula for getting the magnitude and direction of a vector.
Students might be confused between the relationship , which shows the addition of vectors and which shows the addition of magnitudes of vectors.
Suppose, for example, that is the vector representing the total displacement of the person walking in a city, as illustrated in Figure 5.20.
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.
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 Figure 5.21, we use the following relationships:
[BL][OL][AL] Demonstrate a problem involving displacement by physically walking along the specified direction. Show how this can be represented on a graph. Explain that even when solving problems analytically; representing it on a graph would make it easier to visualize the problem.
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 Figure 5.22.
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 .
- 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 Figure 5.23, 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).
- Find the x component of the resultant by adding the x component of the vectors
and find the y component of the resultant (as illustrated in Figure 5.24) by adding the y component of the vectors.
Now that we know the components of we can find its magnitude and direction.
- To get the magnitude of the resultant R, use the Pythagorean theorem.
- To get the direction of the resultant
Classifying Vectors and Quantities Example
This video contrasts and compares three vectors in terms of their magnitudes, positions, and directions.
- . All of them will cancel each other out.
- . Two of them will cancel each other out.
- . Two of them will add together to give the resultant.
- units. All of them will add together to give the resultant.
In the video, the vectors were represented with an arrow above them rather than in bold. This is a common notation in math classes.
Using the Analytical Method of Vector Addition and Subtraction to Solve Problems
Figure 5.25 uses the analytical method to add vectors.
Add the vector to the vector shown in Figure 5.25, using the steps above. The x-axis is along the east–west direction, and the y-axis is along the north–south directions. A person first walks in a direction north of east, represented by vector The person then walks in a direction north of east, represented by vector
The components of and along the x- and y-axes represent walking due east and due north to get to the same ending point. We will solve for these components and then add them in the x-direction and y-direction to find the resultant.
This example shows vector addition using the analytical method. Vector subtraction using the analytical method is very similar. It is just the addition of a negative vector. That is, . The components of – are the negatives of the components of . Therefore, the x- and y-components of the resultant are
and the rest of the method outlined above is identical to that for addition.
What is the magnitude of a vector whose x-component is 4 cm and whose y-component is 3 cm?
- 1 cm
- 5 cm
- 7 cm
- 25 cm
What is the magnitude of a vector that makes an angle of 30° to the horizontal and whose x-component is 3 units?
- 2.61 units
- 3.00 units
- 3.46 units
- 6.00 units
Check Your Understanding
Between the analytical and graphical methods of vector additions, which is more accurate? Why?
- The analytical method is less accurate than the graphical method, because the former involves geometry and trigonometry.
- The analytical method is more accurate than the graphical method, because the latter involves some extensive calculations.
- The analytical method is less accurate than the graphical method, because the former includes drawing all figures to the right scale.
- The analytical method is more accurate than the graphical method, because the latter is limited by the precision of the drawing.
What is a component of a two dimensional vector?
- A component is a piece of a vector that points in either the x or y direction.
- A component is a piece of a vector that has half of the magnitude of the original vector.
- A component is a piece of a vector that points in the direction opposite to the original vector.
- A component is a piece of a vector that points in the same direction as original vector but with double of its magnitude.
How can we determine the magnitude of a vector if we know the magnitudes of its components?
Use the Check Your Understanding questions to assess whether students achieve the learning objectives for this section. If students are struggling with a specific objective, the Check Your Understanding will help identify which objective is causing the problem and direct students to the relevant content.