A. The vector v: 3,1 B. The vector m : 4i 2j C. The vector AB given point A( 2, 2) and B(1, 6)

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2 1. Sec 7.1 Basic Vector Forms Name: 1. Describe each of the following vectors in rectangular form x, y and using component vectors i and j (e.g. 3i + 4j) A. Describe Vector AB : B. Describe Vector CD : C. Describe Vector EF: D. Describe Vector GH : 2. Describe each of the following vectors in rectangular form x, y and using component vectors i and j (e.g. 3i + 4j) A. Describe Vector MN given B. Describe Vector ST given point M( 4, 2) and N(2, 6) point S(3, 2) and T ( 4, 6) 3. Create a graph of each of the following vectors in standard form. A. The vector v: 3,1 B. The vector m : 4i 2j C. The vector AB given point A( 2, 2) and B(1, 6) M. Winking Unit 7-1 pg. 123

3 4. Describe each of the following vectors in polar form r, θ. A. Describe Vector p: B. Describe Vector q: C. Describe Vector m : 5. Create a graph of each of the following vectors in standard form. A. The vector v: 4,150 B. The vector u : 3, 140 C. The vector p: 2,60 6. Determine at least 4 ways to describe vector q in polar form: M. Winking Unit 7-1 pg. 124

4 7. Rewrite each of the following vectors from polar form to rectangular form. A. Rewrite vector m in rectangular form and graph it on the rectangular graph paper in standard position. B. Rewrite vector n in rectangular form and graph it on the rectangular graph paper in standard position. C. Rewrite vector p in rectangular form and graph it on the rectangular graph paper in standard position. 8. Consider the golfer below. He struck the ball so that it was moving at a speed of 200 feet per second at a 70 angle. A. Write the vector in polar form. B. Write the vector in rectangular form. 70 C. If we assumed that the green was 800 feet away horizontally, how long would it take the golf ball to reach the green (assuming air resistance wasn t a factor)? M. Winking Unit 7-1 pg. 125

5 9. Consider the ramp shown below. A car that weighs 2300 pounds is being pushed up a ramp with a 9 elevation. How many pounds of force must the people use just to hold the car in place on the ramp? 10. Rewrite each of the following vectors from rectangular form to polar form. A. Rewrite vector AB in polar form and graph it on the polar graph paper in standard position. B. Rewrite vector u in polar form and graph it on the polar graph paper in standard position. C. Rewrite vector CD in polar form and graph it on the polar graph paper in standard position. M. Winking Unit 7-1 pg. 126

6 6 fps Finding Theta (where α is the reference angle) Quad 1 (x & y positive) e.g. v : 3, 2 Quad 2 (x negative, y positive) e.g. u : 2, 2 Quad 3 (x & y negative) e.g. p: 2, 3 Quad 4 (x negative, y positive) e.g. q: 1, 3 tan 1 ( 2 ) = α tan 1 ( 2 ) = α = 45 2 tan 1 ( 3 ) = α tan 1 ( 3 ) = α θ θ = = 135 θ = θ = = α = 180 α = α = 360 α 11. Perform the following vector operations. A. Consider m : 7,4 determine m. B. Rewrite vector n : 9, 260 in rectangular form. 12. A balloon is floating up in to the sky at a rate of 6 feet per second. At the same time the wind is blowing the balloon horizontally due east at a speed of 10 feet per second. A. How fast is the balloon actually moving? B. What is the angle of elevation of the balloon s ascent? 13. A barge full of containers is being moved across a bay. One tow boat is pulling the barge due North with a force of 5000 Newtons. A second tow boat is pulling the barge due West with a force of 3800 Newtons. A. In what direction will the barge move? B. How much force is being applied in that directions? M. Winking Unit 7-1 pg. 127

7 1. Sec 7.2 Basic Vector Operations Name: 1. Create the following vector statement in the graph to determine the resultant vector in rectangular form. A. Using the information below graphically determine p + j + 2m Show work here B. Algebraically find p j 2m m: p : j : 2 m : 1a. These should be the same 1b. C. Using the information below graphically determine k 3q + 2n Show work here D. Algebraically find k 3q 2n k : 1c. q : n : 3 q : 2 n : These should be the same 1d. M. Winking Unit 7-2 pg. 128

8 2. Let the following vectors be defined: v : 3, 5 u : 7, 2 p : 4, 9 A. Simplify the following vector expression v + 2u and write your answer in component form. B. Simplify the following vector expression p v (i.e. What is the magnitude of the resultant vector?) 2a. 2b. C. Find the direction of the vector expression v + u + p. D. Simplify the following vector expression 2u + p and write your answer in polar form. 2c. 2d. 3. Let the following vectors be defined: p : 4, 30 q : 3, 120 A. Rewrite vector p in rectangular form. 3a. B. Rewrite vector q in rectangular form. 3b. C. Determine p + q in rectangular form. 3c. D. Determine p + q in polar form. 3d. M. Winking Unit 7-2 pg. 129

9 4. Let the following vectors be defined: p : 4, 30 q : 3, 120 Determine p + q in polar form. (Use law of Cosines to determine your answer.) Let the following vectors be defined: a : 2, 40 b : 4, 340 A. Rewrite vector a in rectangular form. 5a. B. Rewrite vector b in rectangular form. 5b. C. Determine b a in rectangular form. 5c. D. Determine b a in polar form. 5d. E. Determine 5a in polar form. 5e. 6. Let the following vectors be defined: m : 3, 134 n : 6, 33 Determine m + n in polar form. (Use law of Cosines to determine your answer.) 6. M. Winking Unit 7-2 pg. 130

10 7. Let the following vectors be defined: v : 6, 38 u : 2, 163 Sketch the vector expression v + 2u in polar form. Start the vector sketch with the point shown below. 8. Sketch a vector graph of two movers trying to move a washing machine up a set of stairs. The first mover is pushing at an angle of 165 and a force of 140 pounds. The second mover is pulling at an angle of 105 and a force of 130 pounds. What is the estimated magnitude and direction of the resultant force based on your drawing? If the movers require a minimum of 200 pounds of force at roughly a 135 angle, do you think they will have enough force to move the washing machine up the stairs? M. Winking Unit 7-2 pg. 131

11 9. A boy is swimming across a river. The boy s path makes a 36 angle with the river bank and the boy is swimming slightly upstream (against the current). The boy is swimming at a rate of 4 feet per second and the current is flowing at 2 feet per second. How fast is the boy actually moving and in what direction? 36 If the river is roughly 31 feet across in width, how many seconds will it take the boy to get across the river? 10. Two tow trucks are trying to pull a car out of the mud at the same time. The first tow truck is pulling the car due East with a force of 900 Newtons. The second truck is pulling the same car 34 North of East from the first tow truck with a force of 1400 Newtons. Which direction will the car most likely move and how many Newtons is being applied to the car? M. Winking Unit 7-2 pg. 132

12 1. Sec 7.3 Vector Matrices and Transforms Name: 1. Let the following vectors be defined: a : 3, 5 b : 7, 2 c : 4, 9 d : 2, 1 A. Rewrite each vector as a column & row matrix. Then, store each as a column matrix in your Graphing Calculator. (Press MATRIX EDIT and change the dimensions to 2 x 1 and enter the vector components) B. Using your Graphing Calculator evaluate the following: (i) a 3b (ii) 4d (iii) 2b + 3a 4c e.g. 2. Using matrices to define vectors can be helpful to create transformations of vectors. Graph each matrix expression as a vector in standard position and describe how each vector compares to vector v. A. The vector v: [ 3 1 ] B. The vector u : [ ] [ 3 0 ] C. The vector w : [ ] [ 3 1 ] M. Winking Unit 7-3 pg. 133

13 Here are the 2-dimensional vector transformation matrices. Reflection Matrices Reflect over y-axis Reflect over x-axis Reflect over y = x Reflect over y = x Rotation Matrices about the Origin Rotate by 90 Rotate by 180 Rotate by 270 Rotate by θ cos sin sin cos 3. Given that vector v can be defined as v : [ a ] describe what transformations take place with each of the b following (compared to v ) A. The vector p: 3 [ ] [ a 0 ] B. The vector q: [4 b 0 4 ] [ a b ] 4. Given that vector u can be defined as u : [ 2 ], perform the following transformations on the vector. 6 A. Rotate vector u 180 about the origin and decrease the magnitude by ½. B. Reflect vector u over the y axis. Then, rotate 50 about the origin and finally, dilate it by a factor of 2. M. Winking Unit 7-3 pg. 134

14 1. Sec Dimensional Vectors Name: 1. Determine the rectangular form of the 3 dimensional vector AB. Given that the points A and B are defined as A: (2, 3, 4) and B: (4, 5, 5) as shown in the graph. B A 2. Find the rectangular form of the vector from point A to point B. a. A( 3, 2, 5) B(6, 2, 2) b. A( 1, 3, 0) B(8, 3, 1) c. A( 4, 2, 7) B(4, 4, 6) 3. Sketch a graph of the 3 dimensional vector shown below in standard position: v: 2, 4, 2 M. Winking Unit 7-4 pg. 135

15 4. Describe the vector CD in standard position at the right using component vectors i, j, k. z 2 CD Find the magnitude of the vector in standard position. x 2 5 y Describe the vector GH in standard position at the right using component vectors i, j, k. 2 z GH Find the magnitude of the vector in standard position y -2 x 8. Using algebra find the resultant vector of CD GH from problems #4 and #6. 9. Given the vectors v 2,1,5 and w 3, 2, 4 find the following: a. 2v 3w b. 3v w c. 2v 4w M. Winking Unit 7-4 pg. 136

16 10. Determine the spherical form of the 3 dimensional vector v: 2, 2,4 11. Determine the spherical form of the 3 dimensional vector u : 4, 3, 2 M. Winking Unit 7-4 pg. 137

17 3 mph 12. A satellite is located in space is orbiting space about 250 miles above Maine. The monitoring station in Houston, Texas can tell the satellite is 1450 miles due East, 1280 miles due North, and 250 miles above the Earth. If we simplify the problem and consider all measures given to be linear (i.e. assuming the curvature of the Earth wasn t used to determine the distances), what is the distance in miles from the monitoring station to the satellite? Can you determine the azimuthal angle (i.e. horizontal angle) using degrees East of North using the monitoring station as the origin? Can you determine the angle of elevation the satellite dish should be set to in order to aim at the satellite? 13. A blimp air ship is ascending vertically at 3 miles per hour due to its buoyancy. The ship is also being propelled by a propeller at 25 miles per hour due East and a wind is blowing due South at a rate of 5 miles per hour. How fast is the air ship actually moving? UP 25 mph E S Can you determine the azimuthal angle heading (i.e. horizontal angle) using degrees East of North using the blimp as the origin? M. Winking Unit 7-4 pg. 137

18 1. Sec 7.5 Advanced Vector Operations Name: The dot product of two vectors can be calculated two ways: u v = u v cos(θ) Magnitude of vector u Magnitude of vector v Cosine of the angle,, between the two vectors The same value can also be determined by finding the sum of the products of each corresponding element. u v = u 1 v 1 + u 2 v u n v n For example, let u : 4,6 and v: 3,2 as shown at the right. u v = ( 4)(3) + (6)(2) = = 0 Notice that the if the dot product of two non-zero vectors is 0 then the two vectors are perpendicular. This could be proven to some extent for a 2-dimensional vectors without too much difficulty if we can assume that perpendicular lines have negative reciprocal slopes. Consider vector a: x, y would have a slope of y and a vector that is perpendicular must then have a slope x of cx where c is some constant, which would suggest that a perpendicular vector would be of the form cy b : cy, cx which would have a dot product of a b = (x)( cy) + (y)(cx) = cxy + cxy = 0 1. Find the dot product of the following set of vectors, graph the vectors, and determine which are perpendicular. a. p: 4,3 c. u : 4, 3,2 and v: 3,2, 3 and q: 4,5 b. m : 8,2 and n : 1, 4 M. Winking Unit 7-5 pg. 138

19 2. Answer the following using the dot product. a. Consider the vectors p: 6,4 and q: 6, a. Using the dot product, what value of a would ensure the vectors are perpendicular? b. Consider the vectors m : 2, 3, 1 and n : 5, b, c. Using the dot product, determine 2 sets of values that would ensure the vectors are perpendicular. (Can you determine a general solution to ensure the vectors are perpendicular?) The cross product of two 3-d vectors can be calculated two ways: u v = ( u v sin(θ)) n Magnitude of vector u Magnitude of vector v Sine of the angle,, between the two vectors The same value can also be determined by finding the determinant of the matrix created by i j k u v = det u 1 u 2 u 3 v 1 v 2 v 3 The unit vector at right angles to both vectors For example, let u : 1, 2, 1 and v : 2, 1, 1 as shown at the right. u v = det i j k = i j k i j ( 2i + 2j 1k ) (1j 1i 4k ) = 1i + 1j + 3k = 1,1,3 = c This cross product creates a new vector that is perpendicular to both of the original vectors simultaneously which we could verify with the dot product: u c = 1, 2, 1 1, 1, 3 = (1)( 1) + ( 2)(1) + (1)(3) = = 0 v c = 2, 1, 1 1, 1, 3 = (2)( 1) + ( 1)(1) + (1)(3) = = 0 Since the dot products are each zero that suggests that each of the two sets of vectors are perpendicular but the dot product of u v = 5 so the original vectors u and v are not perpendicular M. Winking Unit 7-5 pg. 139

20 3. Find the cross product of the following set of vectors a. m : 3, 4,2 and n : 5, 2, 1 b. p: 3, 1,2 and q: 6, 2, 4 4. Consider the vectors u : 3,2,2 and v: 4, 1, 2 a. Determine u v b. Determine v u c. Is the cross product commutative? The angle between two vectors can be calculated using the Law of Cosines: c 2 = a 2 + b 2 2ab cos(c) M. Winking Unit 7-5 pg. 140

21 The formula can be extended to higher dimensions as well. Using the formula complete the problems below. cos(θ) = u v u v 5. Graph the following vectors and find the angle between the following set of vectors a. a: 4, 2 and b : 1,4 b. m : 6, 4 and n : 2,7 c. u : 2, 2, 3 and v: 3, 2, 1 M. Winking Unit 7-5 pg. 141

22 1. Sec 7.6 Complex Numbers (Review) Name: For possibly several well over a thousand years, mathematicians disregarded the idea of attempting to find the square root of a negative number because at the time they thought it was absurd and useless. There is no real number that when multiplied by itself is a negative value which makes finding something like 4 seemingly impossible. 2 2 = = = = 1 According to some references the first written suggestion of attempting to find the square root of a negative number may have dated all the way back to 50 A.D when Heron of Alexandria was trying to determine the volume of an impossible section of the pyramid. The first substantial noted works about finding the square root of a negative number didn t appear again until the 1500 s when there was a math duel to see who could solve general cubic equations more effectively between three Italian mathematicians, Cardano and Ferrari versus Tartaglia. Ferrari was a student of Cardono and stood in for him during the duel. Cardano eventually published these findings in the Ars Magna. There was some argument as to who was first and who was better but the end result was that the solutions required taking square roots of negative numbers. Later in the 1600 s it was Rene Descartes, considered the father of analytical geometry, that accidentally coined the term imaginary to represent the number 1 as well as the standard form for complex numbers of a + b 1. Finally, it was Leonhard Euler that gave us the standard notation of i to represent the number 1 and finalize the complex notation we use today of a + bi. Today, the understanding of imaginary numbers are commonly used in several engineering studies of such topics as force stresses, electrical engineering, and resonance. 1. Find all square roots of each of the following and circle the principal square root. a. 196 b. 1 c. 0 d. 1 e What does i represent? Which mathematician was the first to call the number imaginary? What is standard form? Which mathematician was the first to use the symbol i? What do you think i 2 should be? Can you explain why it MUST be defined? 3. Simplify: a. i 3 = b. i 4 = c. i 11 = d. i 100 = e. i 103 = f. i 131 = M. Winking Unit 7-6 pg. 142

23 4. Simplify the following: a. 45 b Add/Sub and simplifying the following and write the answer in standard form (a + bi): a. 2i 5i b. 3 5i 2 6i c. 2 3i 8 2i 6. Add/Sub and simplifying the following and write the answer in standard form (a + bi): a. 3i b Mult/Div and simplify the following: a. 2i 5i c d. i 15 i 20 e. 2 4i7 8i f. 3 4i 2 g. 3 5i 3 5i M. Winking Unit 7-6 pg. 143

24 8. What are the complex conjugates of each of the following? a. b. 4 6i 8i c Mult/Div and simplify the following: b. 10i 6i b. 2 5i 3i c. 6 5i 4 i d. 3 2i 5 3i M. Winking Unit 7-6 pg. 144

25 10. Mult/Div and simplify the following: a. 42i 2i 3 b. 6i 2 3i 3 3i 11. Solve the following complex equations a. 8i 12 4xi 2y b i i x i y c i x y i M. Winking Unit 7-6 pg. 145

26 1. Sec 7.7 Advanced Vector Operations Name: The complex plane is a graphical interpretation of the complex numbers. The x-axis becomes the REAL NUMBER axis and the y-axis becomes the IMAGINARY NUMBER axis. Using this concept, we can plot complex numbers using a rectangular Cartesian coordinate system. Consider the complex numbers a = 6 + 2i b = 2 5i We can find their sum by using a vector style approach. a + b = (6 + 2) + (2 + 5)i = 4 3i We can also use a graphical head to tail vector style approach: The resultant vector is located at 4 3i 1. Plot the following complex numbers or show the suggested expression. A. 4 3i B. 4i C. c = 3 + 2i & d = 5 + 2i c + d M. Winking Unit 7-7 pg. 142

27 2. Rewrite each of the following complex numbers in polar form and graph. A. Rewrite the complex number 4 + 3i in polar (cis) form and re-graph it on the polar graph paper. B. Rewrite the complex number 4 + 2i in polar (cis) form and re-graph it on the polar graph paper. 3. Rewrite each of the following complex numbers in rectangular form and graph. A. Rewrite the complex number 4(cos i sin 250 ) in rectangular form and re-graph it. B. Rewrite the complex number 3 cis 320 in rectangular form and re-graph it M. Winking Unit 7-7 pg. 143

28 4. Determine the following. A. Evaluate 7 + 4i B. Find the argument of 5 3i C. Find the modulus of 12 5i D. Find the exact rectangular form of 4 cis Consider the complex numbers a = 3 cis 60 and b = 2 cis 150 A. Graph each complex number. B. Rewrite each complex number in approximate rectangular form. C. Algebraically, multiply the two complex numbers in rectangular form. D. Rewrite the product ab in polar form. E. Graph the product of ab. M. Winking Unit 7-7 pg. 144

29 6. Using this Evaluate the following and leave your answer in polar form check your answers with your calculator. A. Evaluate (4 cis 100 )(3 cis 220 ) B. Evaluate ( 2 cis 230 )( 18 cis 200 ) C. Evaluate (7 cis 130 ) 2 D. Evaluate (20 cis 260 ) (5 cis 150 ) 3 E. Evaluate 25 cis 250 F. Evaluate 8 cis 360 G. Evaluate (8 cis 70 ) + (2 cis 210 ) M. Winking Unit 7-7 pg. 145