SB Ch 6 May 15, 2014
Warm Up
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SB Ch 6 May 15, 2014
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
a,b
Recall: direction is measured from the positive x­axis counterclockwise
Bearing (aka heading) is measure from due north clockwise
Component form of a vector
Standard Form is the vector from the origin to the point (a,b)
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HMT (head minus tail) Rule: an arrow given initial point (x1,y1) and end point (x2,y2) represents the vector x2 ­ x1 , y2 ­ y1 .
Example: Show that RS & OP
are equivalent vectors.
Magnitude: denoted v can be found by:
v =
Example:
(x2 ­ x1)2 + (y2 ­ y1)2
If v = a , b
then v =
a2+b2
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Example:
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Example:
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Day 1 Homework Page 464 #1-24 mod 3
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Day 2
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Component Form
Unit Vector Form
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Example:
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Example:
To find magnitude use:
To find direction use:
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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).
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Day 2 Homework: Page 464 #29, 32, 34, 35, 37, 43, 45, 46, 49
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Quick Review
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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.
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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.
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Section 6.2 Dot Product of Vectors
Properties
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EXAMPLE:
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Example:
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Homework Section 6.2
Page 472 #1-22 mod 3, 43, 44, 61-64
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SKIP
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SKIP
Vectors are parallel is u = kv for some constant k.
Example: Prove that the following vectors are orthogonal
2,3
&
-6,4
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DAY 2
Warm Up: Find the dot product.
SKIP
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SKIP
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Homework Section 6.2 Day 2
Page 473 #25-31, 39-51 multiples of 3, 61-66
SKIP
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Warm Up
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DAY 1
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Day 1 Homework: Page 482 #1-10, 11-25 odds
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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 x1 = t and y1 = _____________. (think of this as x
being the time, and y being the height with respect to time)
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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(t3 - 20t2 + 110t - 85)
where 0 ≤ t ≤ 12
Use parametric equations and a calculator to simulate his motion. Estimate the
times when Gary changes direction.
Answer: x1 = ___________ and choose y1 = 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, x1 stays the same for x2, however,
y's equation would change to y2 = _____. Trace your graph to see where the spots are that Gary
changes direction.
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Notes: Initial velocity can be represented by the vector
v = <vocosθt, vosinθ>
Path of the object modeled by parametic equations:
x = (vocosθ)t &
y = -16t2 + (vosinθ)t + yo
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)
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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
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SB Ch 6 1. Find the dot product of
May 15, 2014
<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.
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Warm Up
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(this is notation for showing all solutions (not just 0-2π))
Converting between Polar and Rectangular Coordinates
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Examples
*Remember to check for the
quadrants where tanθ is positive vs
negative.
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Examples
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Section 6.4 Homework
Page 492 # 1-30 mod 3; 43- 49 odd, 51, 52
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Warm Up
3x + 4y = 2
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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 sinθ
r = 2 + 3 cosθ
r = 2 ­ 3 cosθ
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r = 4 + 1 cos θ
r = 2 ­ 2 sin θ
r = 1 ­ 4 sin θ
r = 3 + 2 cos θ
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Spiral Graphs
r=θ
windows:
θ: 0-1440 by 6
x & y: -1000 - 1000 by 100
*changing window settings will alter how many spirals
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Homework Section 6.5
Page 500 #7 ­ 12 & 61 ­ 66
pg 493 #55­60
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Warm Up Answers
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Example:
(exact values)
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Example:
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Homework Day 1: Page 511 #1­30 odds
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SKIP DAY 2
THIS SECTION
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SKIP
Example:
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SKIP
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SKIP
Example:
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Homework Day 2: Page 511 #31­56 odds, 67­70
SKIP
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