Math 144 Activity #10 Applications of Vectors

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Wilfrid Wright
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1 144 p 1 Math 144 Actiity #10 Applications of Vectors In the last actiity, yo were introdced to ectors. In this actiity yo will look at some of the applications of ectors. Let the position ector = a, b and = c, d + Each of these ectors has both a horizontal and ertical component. Each of the components are drawn below. = x, y + = x, y In order to add expressions in algebra, yo can only combine like terms. When it comes to ectors, yo combine like components, so yo end p with the following pictre. y The resltant ector wold be eqialent to x+ x, y+ y. Draw the resltant ector in y the pictre at the left. x x Mathematically yo get: + = x, y + x, y = x+ x, y+ y. Do yo get the same resltant ector if yo do +? Explain why or why not.

2 144 p 2 Find the resltant ector for + w; then draw, w, and + w on the graph below. Label each of the ectors on the graph. = 0, 6, w = 5, 2 Now think abot sbtraction. Is this going to work in the same way? Try it with = x, y and = x, y. Find. Start by explaining what represents. Find. Is it the same as? Explain yor answer. Find the resltant ectors for w and w ; then draw, w, w, w, and w on the graph below. Be sre to label each of the ectors on the graph. = 1, 3, w = 4, 5

3 144 p 3 How are w and w similar? How are they different? Dot Prodct: Piece by Piece Let s explore the idea of dot prodct. Like mltiplying complex nmbers, we want to see how each horizontal and ertical component interacts. To do this, let s break down the ectors sing pictres x x y x x y y y Since the middle two qantities each hae a horizontal (x) and ertical (y) component inoled, what shold their ales be? Explain yor answer. Think abot in terms of filling a bcket with water. If the bcket is held sideways (horizontally) nder a stream of water flowing (ertically), how mch water will end p in the bcket?

4 144 p 4 The dot prodct shold then be eqal to the real nmber fond by taking the sm of the corresponding component prodcts, which is mathematically represented as the following: = x x + y y Find each of the following dot prodcts. 1. p = 5, 2 and q = 3, 7 2. p = 3, 6 and q = 2, 5 3. p = 7 2, 3 and q = 2 2, 9 Dot prodct can also be calclated a second way. Consider the ectors in the diagram below, y - If yo apply the law of cosines to the triangle pictred, making sre to se the magnitde of each ector as the length of each side, yo get 2 = cos x then sing algebra and properties of ectors yo will end p with = cos which is eqialent to both of following: cos cos 1 Explain why they are eqialent. Using this formla for dot prodct will allow yo to be able to find the ale of the angle between the two ectors.

5 144 p 5 Use this formla to find the angle between the ectors listed. 1. p = 5, 2 and q = 3, 7 2. p = 3, 6 and q = 2, 5 3. p = 7 2, 3 and q = 2 2, 9 Frther Applications of ectors: What is work? To gie a mathematical definition, the amont of work done is the component of force along a distance D times the distance the object is moed. This wold gie s a formla of W = F cos D. Now if the force is measred in ponds and distance is measred in feet, what are the nits associated with work? Use what yo hae learned abot ectors and work together as a grop to answer the following qestions. Draw pictres to show the action occrring in each problem. Be sre to inclde the correct nits for each of yor answers. 1. An old-time farmer is plowing his field with a mle. How mch work does the mle do in plowing one length of a field 300 feet long, if it plls the plow with a constant force of 250 lbs and the straps make a 30 0 angle with the horizontal. 2. As part of a togh-man contest, participants are reqired to pll a bs along a leel street for 100 feet. If one contestant did 45,000 ft-lb of work to accomplish the task and the straps sed made an angle of 5 0 with the street, find the tension in the strap dring the pll.

6 144 p 6 3. A home owner applies a force of 40 N to psh her lawn mower back and forth across the back yard. Find the amont of work done if the yard is 50 m long, reqires 24 passes to get the lawn mowed, and the mower arm makes an angle of 39 0 with the leel grond. 4. To break p the monotony of a long, hot, boring Satrday, a father decides to gie his kids a ride in a wheelbarrow. He applies a force of 30 N to moe the load 100 m, then stops to rest. Find the amont of work done if the wheel barrow makes an angle of 20 0 with leel grond while in motion. 5. An old-time farmer is plowing his field with a mle. The length of the farmer s field is 175 feet long and the mle plls the plow with a constant force of 225 lbs. If the work done by the mle is 175 ft-lbs, what is the angle at which the force is being applied?

6.4 VECTORS AND DOT PRODUCTS

458 Chapter 6 Additional Topics in Trigonometry 6.4 VECTORS AND DOT PRODUCTS What yo shold learn ind the dot prodct of two ectors and se the properties of the dot prodct. ind the angle between two ectors