Welcome to IB Math - Standard Level Year 2 Why math? Some things to know: www.aleimath.blogspot.com 1. Lots of info at 2. HW yup. You know you love it! Be prepared to present. Notebook all work is in it. Be prepared to turn in your notebook for evaluation! 3. Content: 4. Grading Ultimately, you need to pass the IB exam! Presentations & HW (20%), quizzes, tests (60%), Exploration (20%) 5. Bring: Notebooks Mandatory! ($3!), pencil(s), calculator, and you! 6. Let's look at the plan in more detail...course overviews sign, parents sign, return! 7. Web page tour: 8. Pass out books 8/30 Welcome 1
What is mathematics? Write one paragraph...be as thorough and clear as you can be. Why learn mathematics? http://www.fordham.edu/info/20603/what_is_mathematics Handout Lockhart's Lament 8/30 What is math? 2
SL QB Practice HL QB Practice Vector Unit Plan 8/30 Vector Syllabus 3
12A.1: #1 3 all (Introduction) 12A.2: #1 3 all (Intro) This summer I went to a math conference in Phoenix. I flew from Santa Fe to Albuquerque. From there, I went on to Phoenix. Draw a diagram that illustrates the path I took. Also show the path I could have taken if I had flown directly from Santa Fe to Phoenix. Eric drove off the road and got stuck in the mud. He starts pushing from the front of the car with a force of 200 pounds. His date, Emily, stands next to him and lifts straight up on the bumper with a force of 100 pounds. Draw a side view diagram that illustrates the forces on the car. Imagine the "total" force being applied to the care by both Eric and Emily. Can you illustrate that in your diagram? Do you notice anything mathematically special about the above situations? They both include ideas that involve both an amount and a direction. We need a word for these kinds of quantities. Vectors and Scalars A vector is a quantity that has both magnitude (amount) and direction. A scalar is a quantity that has only magnitude. How do we represent vectors? 1. One way is with a directed line segment, also known as an arrow. The length of the arrow defines the magnitude (remember complex numbers?) The direction of the arrow defines the direction of the vector. Some conventions: The length is either labelled or implied by drawing the vector on a grid with known scale The direction is often given as an angle measured in a given direction from a given reference > Examples: 20 CCW from horizontal, 15 North of West 2. Another way to represent vectors is with an ordered pair that gives the horizonal and vertical components of the vector. For example: (3, 4) represents the vector This is also written [3, 4] or In books, letters that represent vectors are often bolded and italicized, sometimes with arrows above them. For example: v = [3, 4] or Students should indicate a vector with an arrow above it: The starting point of the vector is called the tail, the ending point is the head. Notice that such vectors do not exist at any particular place in space! They simply describe a magnitude and direction of change or displacement from the tail to the head. They are thus called: Displacement Vectors... can be located anywhere in space 3. A third way to represent a vector is as a segment between two points with specified locations. Often the initial point is the origin, known as O. The vector from the origin to point A is written and is called the position vector of point A since it defines a precise position in space where the vector is located. If the originating point is not the origin, we write which represents the vector from A to B. (Notice that this is not the same as the vector...why not?) The vector is called the position vector of B relative to A. Position Vectors... are located at defined points in space... are "relative" to their starting point... are just called "position vectors" when they are relative to the origin Your turn. v and w are vectors. Talk to your neighbor and decide what you think is meant by:» w = v» w = v» v Let's be more specific. Let v be the vector [3, 4]. Find the vector w if:» w = v» w = v The Opposite of a Vector... has coordinates that are the opposite of the original... is parallel to and in the opposite direction of the original 12A.1: #1 3 all (Introduction) 12A.2: #1 3 all (Intro) 8/31 12A Vector terminology 4
12A.1: #1 3 all (Introduction) 12A.2: #1 3 all (Intro) Present #2 verbal,3 12B.1: #1 4 all (Addition) 12B.2: #1 3 all (Subtraction) (Ex 2: #52.3,30.4,12.2,12.3) The diagram above shows the relative locations of Santa Fe, Albuquerque and Phoenix. Label the cities S, A, and P. What do the vectors, and represent? What would be a logical way to mathematically describe the relationship between them? S A P Geometrically Adding Vectors The sum of two vectors a and b is obtained by placing the vectors head to tail and drawing the vector from the available tail to the available head. The sum is sometimes called the resultant vector. Is a + b the same as b + a? Discuss. What would happen if b = a? The zero vector can be written as (0, 0), [0, 0], Let's try a few: 9/1 12B.1 Vector Add 5
How do you think we should define subtraction of two vectors? Geometrically Subtracting Vectors To subtract vector b from vector a (a b) add the opposite of b to a. The triangle ABC is defined by the following information: On grid paper in your notebook, draw an accurate diagram of triangle ABC Write down the vector C D B 5 A 12B.1: #1 4 all (Addition) 12B.2: #1 3 all (Subtraction) (Ex 2: #52.3,30.4,12.2,12.3) 9/1 12B.2 Vector Subtract 6
12B.1: #1 4 all (Addition) 12B.2: #1 3 all (Subtraction) Present #4 (Associative law).3 &.4 1. Understand the relationship between vector diagrams and their associated equations. 2. Understand scaling a vector. 12B.3: #1 2 all (Vector Equations) 12B.4: #1e h,2 5 all (Scalar multiples) Any vector diagram that forms a closed polygon has an associated equation that describes the relationship. For example, can you write an equation that describes: Are there different answers to this? Constructing equations from diagrams Often it's useful to describe a polygon with a vector equation (computer animation!) Pick any vector as the LHS, then find another path from start to finish, subtracting when needed. Scalar Multiples We have already seen that we can scale a vector's length by multiplying by a constant scalar k. That is, if vector u has a length u then ku has a length k times longer or k u. The scaled vector is in the same direction as the original unless... Multiplying by a negative scalar reverses the direction of the vector. Graphically: u 2u We will look at this algebraically later. 12B.3: #1 2 all (Vector Equations) 12B.4: #1e h,2 5 all (Scalar multiples) 9/6 12B.3, B.4 Equations and Scalar multiples 7
12B.3: #1 2 all (Vector Equations) Present #1ef,2b 12B.4: #1e h,2 5 all (Scalar multiples) Present #3,4 1. Represent vectors using unit vectors 12C: #1 5 all (Vectors in planes) 12D: #1 5 all (Magnitude) (Ex 2: #8.8,8.9,9.3,9.4,14.1,14.2) Vectors are often located on a coordinate plane. Consider the point A at (3, 6). The vector is called the position vector of A. To work efficiently with vectors algebraically, we define two special unit vectors. Base Unit Vectors i is a displacement vector of length 1 in the x direction j is a displacement vector of length 1 in the y direction This way we can describe the vector algebraically: Unit vector form Component form We will work more with this algebraic notation soon... 9/8 12C Vectors in the plane 8
1. Find the magnitude of a vector 2. Understand how to create a unit vector in a given direction The magnitude of a vector, u, is it's length and is written as u. You can find the magnitude from the components by using the Pythagorean Theorem: v b a Note: A scalar has a modulus which also refers to it's size or absolute value. A unit vector is any vector with a magnitude of 1. i and j are sometimes called the base unit vectors. Do you understand why? 12C: #1 5 all (Vectors in planes) 12D: #1 5 all (Magnitude) (Ex 2: #8.8,8.9,9.3,9.4,14.1,14.2) 9/8 12D Magnitude 9
12C: #1 5 all (Vectors in planes) Present #2 5 verbally call on people 12D: #1 5 all (Magnitude) Present 1de,2cde,3cde 1. Add two vectors algebraically 2. Find the opposite of a vector algebraically. 3. Subtract two vectors algebraically 4. Scale a vector to a given length 12E: #1efgh,2def,3,4efgh,5,7fghij,8,9 (2D Operations) Ex 2: #16.3,29.10,29.13,30.2,30.5,30.6,30.7,66.1 So far, we have been working mostly graphically. Let's look at vectors algebraically. We have seen many of these ideas already we'll formalize them now. Sum of Two Vectors Negative Vectors Difference of Two Vectors Scalar Multiplication We can do vector algebra using component form notation... Or... using unit vector notation... If a particular form is requested, use it. Otherwise, choose the one you prefer. Be sure to understand both notations. Vector translations: 12E: #1efgh,2def,3,4efgh,5,7fghij,8,9 (2D Operations) Ex 2: #16.3,29.10,29.13,30.2,30.5,30.6,30.7,66.1 9/12 12E Operations with 2D vectors 10
12E: #1efgh,2def,3,4efgh,5,7fghij,8,9 (2D Operations) Present 4h, 7(gh),8,9 others? 1. Find and use properties of vectors between two points. 12F: #1def,2 8 all (Vector between points) 12G: #2,3,6,7,9,12 17 all (3D Vectors) Ex 2: #17.14,29.3 Vector between two points If point A has coordinates (a, b) and point B has coordinates (p, q) then the vector from A to B has components: This can also be thought of a the difference between the position vectors to B and A. Another name for the vector from A to B is: Note that the direction of the subtraction matters! Try a couple: Of course, you are more likely to see these ideas in a context! (Write and solve an equation) 12E: #1efgh,2def,3,4efgh,5,7fghij,8,9 (2D Operations) 12F: #1def,2 8 all (Vector between points) Ex 2: #16.3,29.10,29.13,30.2,30.5,30.6,30.7,66.1 9/13 12 F Vector between points 11
Vector translations: 1. Sketch and find 3D vectors from points 2. Understand and perform basic operations with 3D vector Conceptually, this is no different than in 2D. However, it's much harder to draw and visualize. Note that in 3D, lines that are not parallel do not necessarily intersect! A 3D space has an origin and 3 mutually perpendicular axes. There are various ways to draw them but the positive directions of the axes must be "right handed". y z z is generally used to represent height in physical problems. x y z x The coordinates of points, components of vectors and base unit vectors all get extended to include the z axis: A = (a, b, c) So: v = ai + bj + ck Sketch the points P ( 3, 1, 2) and Q (1, 1, 3). Find and sketch the vector PQ Magnitude in 3D Work with a partner to calculate the length of the vector [2, 3, 6]. Show that your result is correct. Is there an easier way? Magnitude of 3D Vectors The magnitude of vector v = ai + bj + ck is given by: Applications can get more complex use what you know. You do not have to draw 3D to label 3D! Use your algebra, either in component form or in unit vector form. Quite a few problems - get to work on them now, spread your work out over time! 12F: #1def,2 8 all (Vector between points) 12G: #2,3,6,7,9,12 17 all (3D Vectors) 9/13 12G 3D Vectors 12
12F: #1def,2 8 all (Vector between points) Present 8 extend to general case 12G: #2,3,6,7,9,12 17 all (3D Vectors) Present #7,9b,14,15,17 Change to 12F: #1def,2 6,7a,8 (Vector between points), 12G: #7,9,12 17 all (3D Vectors) Quick Quiz Consider the triangle with vertices A(5, 6, 2), B(6, 12, 9) and C(2, 4, 2) 1. Find the vector from B to A. 2. Find the lengths of all three sides. 3. Hence show that the triangle is a right triangle 4. Find the area of the triangle 1. BA = [ 1, 6, 11] A1 2. BA = 158 A1 AC = [ 3, 2, 4] AC = 29 A1A1 BC = [ 4, 8, 7] BC = 129 A1A1 3. ( 29) 2 + ( 129) 2 = ( 158) 2 A1 4. A = ½ 29 129 = ½ 3741 M1A1 1. Rearrange and solve vector equations in 2D and 3D. 12H: #1 15 odd, or 2 14 even, 15 (3D Operations) 12I: #1 10 even (Parallelism) Ex 2: #11.9,11.10,22.6, 30.3 Nothing new conceptually but the applications can be interesting. Show, using components, that vector addition is or is not: 1. Commutative 2. Associative Other properties of vectors follow directly from things we've looked at previously. We can manipulate and solve vector equations using the following: Try these: 9/15 12H 3D Operations 13
1. Understand and apply properties of parallel vectors in 3D What does parallel mean in 2D? How can you extend this idea to 3D? Some applications: 12H: #1 15 odd, or 2 14 even, 15 (3D Operations) 12I: #1 10 even (Parallelism) Ex 2: #11.9,11.10,22.6, 30.3 Quite a few ideas here - spread the work out! 9/15 12I vectors 14
12H: #1 15 odd, or 2 14 even, 15 (3D Operations) Present #9b,10c,11,15 Verbally (Monday) 12I: #1 10 even (Parallelism) Present #6, 8, 10 Quiz covering Chapter 12 this Thursday, 9/22 1. Understand and find scalar products of vectors 2. Find the angle between vectors 12J: #1 23 odd (Dot product) The Desert Outdoor club plans to have a fruit sale (instead of a bake sale) and sell apples for $0.50, bananas for a quarter and bags of cherries for $1. Next Friday they expect to sell 35 apples, 15 bananas and a dozen bags of cherries. How much money will they raise? We have explored adding, subtracting, and scaling vectors. What is meant by multiplying or dividing vectors? Answer: Nothing! But the dot product or scalar product or inner product is a very useful operation. Do not confuse this with the cross product which is written as v x u and is a totally different idea. (We will not explore cross products in this course) Let u = [a, b, c] v = [p, q, r] and w = [k, m, n] (a) Verify that u v = v u (b) What is the significance of u u? (c) What does u v = 0 tell us about u and v? (d) Is it true that u (v + w) = u v + u w for all vectors u, v, and w? (e) Write (u + x) (v + w) without parentheses. (f) If u and v represent sides of a ogram, then u + v and u v represent the diagonals. What does (u + v) (u v) = 0 tell us about the ogram? 9/19 12J Dot product 15
Finally, the dot product illuminates a very interesting idea: We can rearrange this to see even more clearly that: Alternatively, the dot product of two vectors is given by: Dot Product (Scalar Product) (Inner Product) u v = u v cosθ In the HW, work with the meaning of this. Find the angle between the vectors [3, 4] and [12, 5]. Then find the angle between the vectors [3, 4] and [ 12, 5]. Can you draw any general conclusions? θ is acute <=>u v > 0 θ is obtuse <=> u v < 0 We know that u v = 0 when u and v are perpendicular. What is u v if u and v are parallel? Can you show this with components (Hint: Let u = [a, b] and v = k[a, b]) The HW may stretch you. Again, give yourself time, stretch it out, work this one. 12J: #1 23 odd (Dot product) Quiz on Chapter 12: Thursday, 9/22 FYI: Tomorrow, 9/20, we will answer questions and work in class. 12J: #2 22 even (More dot product) Review 12C: #1 15 (Review) 9/19 12J Angle between vectors 16
Present from 12J #1 23 odd 1. Understand and find scalar products of vectors 2. Find the angle between vectors Work on QB in class: Basics: 3,7,9,10 12J: #2 22 even (More dot product) Review 12C: #1 15 (Review) Chapter test: Thu 9/22 The test will not take the whole period. There will also be a brief lesson and some HW for next Monday! Later: 2D Lines: 1,4,5,6,8 3D Lines: 2 9/20 12J Dot Products (cont) & review 17