PreCalculus Notes. MAT 129 Chapter 10: Polar Coordinates; Vectors. David J. Gisch. Department of Mathematics Des Moines Area Community College

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1 PreCalculus Notes MAT 129 Chapter 10: Polar Coordinates; Vectors David J. Gisch Department of Mathematics Des Moines Area Community College October 25, 2011

2 1 Chapter 10 Section 10.1: Polar Coordinates Section 10.3: The Complex Plane; De Moivre s Theorem Section 10.4: Vectors Section 10.5: The Dot Product

3 Section 10.1: Polar Coordinates Summary We will learn about Polar coordinates; convert rectangular coordinates to polar coordinates and vice versa; and transform equations from polar to rectangular form.

4 Radians vs. Degrees

5 Polar Coordinates Figure: Graph of polar coordinates (2, π/4)

6 Example Graph the polar coordinates ( 4, π ) (, 2, 5π 2 4 ), ( 3, 2π 3 ),

7 Several Forms of a Single Point

8 Conversion from polar Coordinates to Rectangular

9 Transform Polar Equations to Rectangular Form Identities r 2 = x 2 + y 2 x = r cos θ y = r sin θ Example Change r = 4 cos θ to rectangular form.

10 Example Change r = 4 1 cos θ to rectangular form.

11 Example Change y = 3x + 2 to polar form.

12 Example Change x 2 + y 2 = 1 to polar form.

13 Section 10.3: The Complex Plane; De Moivre s Theorem Summary We will plot polar coordinates in the complex plane; find products and quotients of complex numbers; and use De Moirve s theorem to find roots of complex numbers.

14 Definition We can write a complex number a + bi as z = x + yi. The magnitude or modulus of z, denoted as z, is the distance of z from the origin In other words, z = r. z = x 2 + y 2 Note Recall that any complex number a + bi has the conjugate a bi. Thus, any complex number z = x + yi has the conjugate z = x yi.

15 Definition If r 0 and 0 θ 2π, the complex number z = x + yi may be written in polar form as z = x + yi = r cos θ + (r sin θ)i = r(cos θ i sin θ) Note If x = r cos θ then cos θ = x/r and likewise sin θ = y/r.

16 Example Write z = 2 i in polar form.

17 Example Write z = 2 + 3i in polar form.

18 Products and Quotients Theorem Let z 1 = r 1 (cos θ 1 i sin θ 1 ) and z 2 = r 2 (cos θ 2 i sin θ 2 ) z 1 z 2 = r 1 r 2 [ cos(θ1 + θ 2 ) i sin(θ 1 + θ 2 ) ] z 1 z 2 = r 1 r 2 [ cos(θ1 θ 2 ) i sin(θ 1 θ 2 ) ]

19 Example Multiply and divide z 1 = 3(cos 30 i sin 30 ) and z 2 = 2(cos 100 i sin 100 ) in polar form.

20 Example Multiply and divide z 1 = 3 + 2i and z 2 = i in polar form.

21 De Moivre s Theorem Theorem Let z = r(cos θ i sin θ), then z n = r n[ cos(nθ) i sin(nθ) ]

22 Example If z = 2(cos 30 i sin 30 ), calculate z 6.

23 Example Calculate (1 + i) 5 and write your answer in a + bi form.

24 Complex Roots Theorem Let w be a complex number and n 2. Then the complex numbers that are solutions to the equation z n = w are called complex nth roots of w. The complex roots can be calculated as Theorem Let w = r(cos θ i sin θ), then [ z k = n r cos where k = 0, 1, 2,..., n 1. ( θ n + 2kπ ) i sin n ( θ n + 2kπ ) ] n

25 Example Find the complex cube roots of w = 1 3i.

26 Section 10.4: Vectors Summary We will plot vectors; convert vectors to various forms; and perform arithmetic of vectors.

27 Line, Segments, Vectors

28 Note Any two vectors are considered to be equal if they have the same magnitude and direction. There exact starting and stopping points need not be the same. For example, all three vectors below are equal.

29 Adding Vectors When you add two vectors you can think of visually as starting the second vector from where the first one left off. We call the black vector v + w the resultant vector. Note: Order does not matter either.

30 Negative Vectors Given a vector v we can also consider v. The vector v has the same magnitude as v but has a direction opposite of v.

31 Adding and Subtracting Vectors (Visually) Figure: u + v

32 Adding and Subtracting Vectors (Visually) Figure: u v

33 Adding and Subtracting Vectors (Visually) Figure: 2 u + v w

34 Scalar Multiples

35 Properties of Scalar Multiples

36 Magnitude Definition The length of a vector is defined as the magnitude, written as v. Note If v = 1, then we call v a unit vector.

37 Algebraic Vector Definition We write a vector in algebraic form as v = x, y, where x and y are called the vector s components. Thus, we also refer to this form as component form.

38 Algebraic Vector Theorem If a vector v starts with the initial point P 1 = (x 1, y 1 ), not necessarily the origin, and terminal point P 2 = (x 2, y 2 ), then v can be written in algebraic from as v = x 2 x 1, y 2 y 1 I often think of this formula as taking the end point minus the start point.

39 Example Given the following points find the vectors P 1 = (2, 3) P 2 = (4, 3) P 3 = ( 1, 5) 1 P 1 P 2 2 P 3 P 2 3 P 1 P 3

40 Another Form!? Definition We also write vectors in algebraic form as v = a, b = ai = bj. Here i symbolizes the x-component and j symbolizes the y-component. [Note that i is not the imaginary i here.]

41 Adding and Subtracting Vectors (Algebraically) Properties If we have two vectors u = ai + bj = a, b and v = ci + dj = c, d, then u + v = (a + c)i + (b + d)j = a + c, b + d u v = (a c)i + (b d)j = a c, b d α u = (αa)i + (αb)j = αa, αb u = a 2 + b 2

42 Example Given the following vectors calculate the sum or difference. u = 2i + 3j v = 4i + 2j 1 u + v 2 3 u v 3 v

43 Unit Vector Theorem For any vector v, the vector u = v v is a unit vector in the same direction as v. Note that this implies v = v u

44 Example Find a unit vector in the same direction as v = 4i 3j. Example Show that v = v u.

45 Magnitude and Direction Vectors, when written as x, y can be thought of as being in rectangular form, though they do not necessarily start at the origin. Vectors can also be described by their magnitude, given by u, and direction θ, measured from the positive horizontal. Thus we can treat vectors as polar coordinates and vice versa, making the arithmetic similar.

46 Applications Example A ball is thrown with an initial speed of 25 miles per hour in a direction that makes an angle of 30 with the positive horizontal. Express the velocity vector v in terms of i and j. What is the initial speed in the horizontal direction? What is the initial speed in the vertical direction?

47 Applications Example A box of supplies that weighs 1200 pounds is suspended by two cables attached to the ceiling as shown. What is the tension on the two cables?

48 Applications Example At a picnic there is a contest, in which hoses are used to shoot water at a beach ball in three different directions. As a result there are three forces acting upon the beach ball. The pressure and position of the hoses can be given as F 1 = 50N at 120 F 2 = 90N at 240 F 3 = 70N at 0 Here N is the unit of force Newtons. What is the resultant vector (i.e. where is the ball going to go)?

49 Section 10.5: The Dot Product Summary We will find the dot product of two vectors; the angle between vectors; and determine whether vectors are parallel or orthogonal.

50 Dot Product Formula If we have two vectors u = ai + bj = a, b and v = ci + dj = c, d, then the dot product of the two vectors is given as u v = ac + bd

51 Example Given the following vectors calculate given. u = 2i + 3j v = 4i + 2j 1 u v 2 3 u v 3 u

52 Properties of the Dot Product Properties If we have two or more vectors, then u v = v u (1) u ( v + w) = u v + u w (2) v v = v 2 (3) 0 u = 0 (4)

53 Law of Cosines u v 2 = u 2 + v 2 2 u v cos θ

54

55 Angle Between Vectors Theorem For any two vector u and v, the angle θ between them can be determined by u v cos θ = u v Note that this gives the angle so that 0 θ π.

56 Example Given the following vectors calculate the angle between them using the dot product. u = 4i 3j v = 2i + 5j

57 Example

58

59 Parallel and Orthogonal Vectors Theorem If two vectors were parallel they would either be going in the same direction or in opposite directions, resulting in θ = 0 or θ = 180, respectively. As cos(0) = 1 and cos(180) = 1 we can say that two vectors are parallel if cos θ = u v u v = ±1 Likewise, two vectors are said to be orthogonal if they form a right angle. As cos(90) = 0, we know two vector are orthogonal if cos θ = u v u v = 0 which happens if u v = 0

60 Example State whether the vectors are parallel, orthogonal, or neither. u = 3i + 8j v = 2i 16 j w = 6i + 12j 3 1 u and v 2 3 u and w

61 Projection Theorem Given two vectors v and w, the projection of v onto w is v 1 = where v 1 is parallel to w. We also have v w v w w = w 2 w w w which is perpendicular to w. v 2 = v v 1

62 Example Given the vectors u = 4i 3j v = 2i + 10j Find the projection of u onto v, u 1, and the orthogonal vector u 2.

63 In elementary physics, the work W done by a constant force F in moving an object from point A to point B is defined as W = (magnitude of force)(distance) = F AB = F AB Example The figure below shows a girl pulling a wagon with a force of 50 pounds. How much work is done in moving the wagon 100 feet if the handle makes an angle of 30 with the ground?

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