Math 144 Activity #9 Introduction to Vectors
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1 144 p 1 Math 144 ctiity #9 Introduction to Vectors Often times you hear people use the words speed and elocity. Is there a difference between the two? If so, what is the difference? Discuss this with your group. The two quantities are in fact different. Speed is considered a scalar quantity while elocity is a ector. What is a ector? In this actiity you will explore what a ector is and what are they used for. Quantities like area, olume, time, and speed are all considered to be scalar because they can be assigned a single numerical alue with a unit of measure. Force, momentum, displacement, and elocity are some examples of ectors since they require more than one quantity to describe their attributes. Each ector quantity needs both magnitude and direction in order to describe them completely. Let s start with a geometric approach to ectors. Look at the three figures below. What do the pictures represent? Now using what you remember from geometry, name each of the figures below mathematically. B B B The figure that it is going to be explored further is the one in the middle. You may hae remembered it as being called a ray, but it can also be called a directed segment. It has a length, or magnitude, that can be measured and an arrow to indicate a direction. For this reason, it can be a referred to as a ector. Vectors can be named using the same notation that is used for rays or using a bold, lower case letter like and u. Since it can be hard to write in bold using a pencil and paper, you will also see the lower case letter with an arrow oer the top. Name each of the ectors below. B D C F G E H
2 144 p 2 What does it mean for quantities to be equal? Does this apply to ectors? If so, gie a definition for equialent ectors? Look back at the picture of the ectors on the first page; are there any ectors that are equal? If so, what are they? Consider the pictures of the ectors below, describe the relationship between them. Write a mathematical equation to show the relationship. u When working with seeral ectors at a time, it can be easier if they are placed in a coordinate plane. The graphical representation will help with analyzing the interaction between the ectors. In the coordinate plane below, there are two ectors. re the two ectors equal? Explain why or why not. How many units ertically are there between the initial and terminal points for each ector? u How many units horizontally are there between the initial and terminal points for each ector? re they the same? What are the lengths of each ector?
3 144 p 3 Vectors and u are in fact equialent and the location of the ector is unimportant. For this reason, a unique and equialent ector can be placed anywhere in the coordinate plane, this is called the position ector. What point would make a good place to put the initial point of the ector? Draw the position ector for ector w. Does the position ector look like something that you hae seen before in class? If so, what? w s mentioned before ectors hae two components, a horizontal component and a ertical component. On the graph below there is a ector drawn whose initial point is at the origin and terminal point is at an arbitrary point, say a, b, in quadrant I. If we call this ector u, then another notation could be u = a, b. y Draw the horizontal component, as a ector, for ector u. Draw the ertical component, as a ector, for ector u. What shape hae you created? x What is the magnitude of the horizontal component? What is the magnitude of the ertical component? Write a formula to calculate the magnitude of ector u, denoted as u? Explain how you know this will work.
4 144 p 4 You should hae a picture that looks like the figure below, a right triangle drawn in quadrant one. y If this is not what your picture looks like go back, figure out what you did wrong and fix it. Be sure to ask for help if you need it. b a x You hae worked with right triangles all semester long, so you should be able to find the angle that is in standard position. What is it? Is there another way to find? Write it down as well. Make sure that eery member of your group understands how and why each of your answers is correct. Find the magnitude for each of the ectors drawn below. Then find the angle, in standard position, for each position ector. w
5 144 p 5 Let s go back to the position ector u = y a, b with the ertical and horizontal components. Since both the horizontal and ertical components are represented as ectors, write each of them as such. Horizontal component as a ector =, Vertical component as a ector =, x Each component now has two components of their own, but in each case one of the components has a alue of zero. When the two components are put tip to tail, you can see they are connected by the position ector that was started with. What do you get if you add the corresponding components together? If you did this correctly, you should hae what is called the resultant ector; which is equialent to ector u drawn aboe. Placing ectors tip to tail in this fashion, will create a resultant ector. What mathematical operation is this process equialent to? Find the resultant ector when ectors and u are placed tip to tail. Does it matter which ector stays at the origin? Explain or show how you came up with your answer. u
6 144 p 6 Draw each of the gien ectors and then find the resultant ector for the sum of each. 1. = 2, 4, w = 1, 3 2. = 0, 6, u = 5, 2
7 144 p 7 If you can add ectors together, can you subtract them? Try it using the ectors w = 3, 2 and = 5, 4. What are some of the things that you hae to be careful of? If you are not sure ask your instructor.
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