Warm Up 1
Chapter 6: Applications of Trig: Vectors Section 6.1 Vectors in a Plane Vector: directed line segment Magnitude is the length of the vector Direction is the angle in which the vector is pointing a,b (a,b) Recall: direction is measured from the positive x axis counterclockwise Bearing (aka heading) is measure from due north clockwise a,b Component form of a vector Standard Form is the vector from the origin to the point (a,b) 2
HMT (head minus tail) Rule: an arrow given initial point (x 1,y 1 ) and end point (x 2,y 2 ) represents the vector x 2 x 1, y 2 y 1. Example: Show that RS & OP are equivalent vectors. Magnitude: denoted v can be found by: v = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Example: If v = a, b then v = a 2 +b 2 3
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Day 1 Homework Page 464 #1-24 mod 3 6
Day 2 7
Component Form Unit Vector Form 8
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Example: To find magnitude use: To find direction use: 10
Example: A DC-10 jet is flying on a bearing of 65 degrees at 500 mph. Find the component form of the velocity of the airplane. Recall bearing is measured differently than direction. Example: A flight is leaving an airport and flying due East. There is a 65 mph wind with bearing 60 degrees. Find the compass heading the plane should follow and determine what the airplane's ground speed will be (assuming speed with no wind is 450 mph). 11
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Day 2 Homework: Page 464 #29, 32, 34, 35, 37, 43, 45, 46, 49 13
Quick Review 14
Example: A flight is leaving an airport and flying due East. There is a 65 mph wind with bearing 60 degrees. Find the compass heading the plane should follow and determine what the airplane's ground speed will be (assuming speed with no wind is 450 mph). Example: An airplane is flying on a compass heading of 170 degrees at 460 mph. A wind is blowing with bearing 200 degrees at 80 mph. 15
A boat is traveling on a bearing of 300 degrees at 350 mph. A current is moving at 85 mph with direction 75 degrees. Find the actual velocity of the boat in unit vector form. Then find the actual speed and direction the boat is traveling. 16
Section 6.2 Dot Product of Vectors Properties 17
EXAMPLE: 18
Example: 19
Homework Section 6.2 Page 472 #1-22 mod 3, 43, 44, 61-64 20
SKIP 21
SKIP Vectors are parallel is u = kv for some constant k. Example: Prove that the following vectors are orthogonal 2,3 & -6,4 22
DAY 2 Warm Up: Find the dot product. SKIP 23
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Homework Section 6.2 Day 2 Page 473 #25-31, 39-51 multiples of 3, 61-66 SKIP 25
Warm Up 26
DAY 1 27
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Day 1 Homework: Page 482 #1-10, 11-25 odds 31
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Example: Projectile Motion A distress flare is shot straight up from a ship's bridge 75 feet above the water with an initial velocity of 76 ft/sec. Graph the flare's height against time, give the height of the flare above water at each time, and simulate the flare's motion for each length of time. a. 1 sec b. 2 sec c. 4 sec d. 5 sec Step 1: State an equation that can be used to model the flare's height above water t-seconds after launch. Step 2: A graph of the flare's height can be found using parametric equations with x 1 = t and y 1 =. (think of this as x being the time, and y being the height with respect to time) 33
Day 2: Section 6.3 Simulating Motion Example: Simulating Horizontal Motion Gary walks along a horizontal line (think of it as a number line) with the coordinate of his position (in meters) given by s = -0.1(t 3-20t 2 + 110t - 85) where 0 t 12 Use parametric equations and a calculator to simulate his motion. Estimate the times when Gary changes direction. Answer: x 1 = and choose y 1 = 5 (to give space to display this motion) As t values increase, notice the x values are. This means that Gary must have changed direction during his walk. To simulate this, x 1 stays the same for x 2, however, y's equation would change to y 2 =. Trace your graph to see where the spots are that Gary changes direction. 34
Notes: Initial velocity can be represented by the vector v = <v o cosθt, v o sinθ> Path of the object modeled by parametic equations: x = (v o cosθ)t & y = -16t 2 + (v o sinθ)t + y o Hitting a Baseball Kevin hits a baseball at 3 ft above the ground with an initial speed of 150 ft/sec at an angle of 18 degrees with the horizontal. Will the ball clear a 20-ft wall that is 400 ft away? (Remember: You need to change up your window settings to get a nice picture) 35
Review: Riding on a Ferris Wheel Example # 10 page 481 in book Homework Section 6.3 cont. Page 482 #31, 37-40, 43, 44, 46, 51, 59-64 36
1. Find the dot product of <3,-5> and <-6, -2>. 2. With the given vectors in #1, find the angle between them. 3. Vector v has magnitude 8 with bearing 70 degrees. Show the component form of vector v. 37
Warm Up 38
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(this is notation for showing all solutions (not just 0-2π)) Converting between Polar and Rectangular Coordinates 40
Examples *Remember to check for the quadrants where tanθ is positive vs negative. 41
Examples 42
Section 6.4 Homework Page 492 # 1-30 mod 3; 43-49 odd, 51, 52 43
Warm Up 3x + 4y = 2 44
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What is the difference between r =a ± b sin θ and r =a ± b cos θ? r = 2 + 3 sinθ r = 2 + 3 cosθ r = 2 3 sinθ r = 2 3 cosθ 46
r = 4 + 1 cos θ r = 1 4 sin θ r = 2 2 sin θ r = 3 + 2 cos θ 47
Spiral Graphs r = θ windows: θ: 0-1440 by 6 x & y: -1000-1000 by 100 *changing window settings will alter how many spirals 48
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Homework Section 6.5 Page 500 #7 12 & 61 66 pg 493 #55 60 50
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Warm Up Answers 52
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Example: (exact values) 56
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Homework Day 1: Page 511 #1 30 odds 59
DAY SKIP 2 THIS SECTION 60
SKIP Example: 61
SKIP 62
SKIP Example: 63
Homework Day 2: Page 511 #31 56 odds, 67 70 SKIP 64
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