Unit 1 Representing and Operations with Vectors. Over the years you have come to accept various mathematical concepts or properties:

Transcription

1 Lesson1.notebook November 27, 2012 Algebra Unit 1 Representing and Operations with Vectors Over the years you have come to accept various mathematical concepts or properties: Communative Property Associative Property Distributive Property Identities Inverse Operations Problem Solving Skills 4 basic steps to effective problem solving: 1) Understand the problem: 2) Think of a plan: 3) Carry out the plan. 4) Reflect on solution:

2 Lesson1.notebook November 27, 2012 Pythagorean Theorem Review of Trigonometry Right Angle Trigonometric Ratios Sine Law Cosine Law

3 Lesson1.notebook November 27, 2012 CAST Rule Example: From a cliff 50 m high, a person can see 2 boats. One boat is S50W at an angle of depression of 30 degrees. The second boat is S60E and is at an angle of depression of 45 degrees. How far apart are the boats?

4 Lesson1.notebook November 27, 2012 Vector Concepts Today's goal: I can explain the difference between a vector and scalar and I can describe the properties and notation associated with vectors. Vectors Scalar Examples: Vector Notation

5 Lesson1.notebook November 27, 2012 Angles Between Vectors Scalar Multiplication Zero Vector Unit Vectors Examples: List all equal vectors in the given diagram.

6 Lesson1.notebook November 27, 2012 Example: For the given number line, state equivalent vectors. A 3 B 2 C 5 D 7 E Homework: pg 121 #1 6, 8, 9 Handout pg 127 #1,2,4,6,8,9,13

7 Lesson2.notebook November 27, 2012 Algebraic Vectors Today's goal: I can write a vector in both ordered pair and unit vector notation and complete the associated operations using algebraic methods. There are two algebraic forms that we can use to represent vectors: Ordered Pair Notation Unit Vector Notation In 3 space: The Direction Cosines for a vector in 3 space are: From our grade 11 trigonometric identity: It now becomes:

8 Lesson2.notebook November 27, 2012 Example: Determine the unit vector notation for the following vectors. A) u = [ 9, 12] B) v = [3, 5, 8] Example: Write the following in order pair notation. A) u = 20, [N20W] B) v = 50, θ = 230

9 Lesson2.notebook November 27, 2012 Example: Determine the magnitude, direction and unit vector for the following: A) u = [2, 6] B) v = [1, 3, 9] For 3 space we can break it into "areas" or "planes". For example: HW page 166 # 1 8, 11, 13, 14, 16 19, 23

10 Lesson3.notebook November 27, 2012 Operations with Algebraic Vectors Today's goal: I can extend my knowledge of algebraic vectors to the concepts of adding and subtracting, making the connection with currently understood algebra rules. Two Algebraic vectors are equal if and only if their Cartesian components are equal. Collinear vectors are any two vectors that lie on the same line. Which means: Operations with Algebraic Vectors Scalar multiplication: k(ai + bj) = k[a, b] = Vector addition/subtraction: ai + bj (ci dj) = Example: If u = [3,8] and v = [ 2, 6], determine: A) w if w = 3u + 2v B) w if w = 2u + 1 / 2 v C) u + 3v

11 Lesson3.notebook November 27, 2012 Example: Given points P( 6, 1), Q( 2, 1) and R( 3, 4), determine: A) QP B) RP Example: Given points P(3, 1, 4), Q( 3, 1, 5) and R( 7, 4, 2), determine: A) QP B) RP

12 Lesson3.notebook November 27, 2012 Example: Determine if the following points are collinear. Justify your answer. P(2, 3, 6) Q(8, 1, 10) R(5, 2, 8) HW page 172 #2 10, 13, 14

13 Lesson4.notebook November 27, 2012 Vector Laws Today's goal: I can apply the rules of vector addition to applications that model real world situations, preparing me to apply them in a physics context. If u and v are vectors: 1) Closure 2) Communative 3) Associative 4) Distributive (if a and b are scalars) 5) Identity Triangle Law of Vector Addition When adding/subtracting two vectors: 1) Draw the two vectors head to tail. 2) The "sum" or resultant is a vector that is measured from the tail of the first vector to the head of the second vector.

14 Lesson4.notebook November 27, 2012 Parallelogram Law of Vector Addition When adding/subtracting two vectors: 1) Draw the vectors tail to tail. 2) Create a parallelogram using the two vectors as two sides of the parallelogram. 3) Draw the resultant vector which extends from the tails to the heads of the vectors or it is the diagonal of the parallelogram. Example: if u = 10 units, v = 5 units and the angle between the two vectors when placed tail to tail is 30 degrees, determine the resultant.

15 Lesson4.notebook November 27, 2012 Geometric Vectors Example: Determine the resultant of 3u 2v + 5w. Algebraic Vectors To use algebraic vectors we need to review the right angle triangle and discuss components.

16 Lesson4.notebook November 27, 2012 Example: if u = 10 units, v = 5 units and the angle between the two vectors when placed tail to tail is 30 degrees, determine the resultant. Algebraic Vectors (Coordinate Form) Example: Determine the resultant of 3u 2v + 5w.

17 Lesson4.notebook November 27, 2012 Determine u given the following vectors u = 25 v = 10 R = 15 Homework: Pg 133, # 1, 3ab, 4 6, 10 12, 20 Handout

18 Lesson6.notebook November 27, 2012 Force and Velocity of a Vector Today's goal: I can apply previous understanding of vector addition and subtraction and apply them to real world situations involving physics. Force For this section everyone needs to "know" Newton's Three Laws: 1) 2) 3) Example: A 5 kg sign is suspended as shown. Determine the tension in each wire

19 Lesson6.notebook November 27, Example: Chris and Josh are pulling a sled as shown. If Josh is pulling with 70 N of force, determine with what force Chris must exert so the sled is kept travelling straight. 70 N x

20 Lesson6.notebook November 27, N x Example: Determine the resulting force.

21 Lesson6.notebook November 27, 2012 Velocity as a Vector Velocity is a measure of speed (magnitude) and direction. We can use our knowledge of vectors to solve questions involving velocity. Relative Velocity is the velocity an observer perceives other objects to be travelling at. Complete the following chart: Bus: Train: Car: My Location Bus Train Car

22 Lesson6.notebook November 27, 2012 Relative Motion Equation: v rel = v A + v B where, v rel v A v B Example: A plane is travelling 400 km/h [N60E]. The pilot set a course of 500 km/h [N30E]. Determine the velocity of the wind.

23 Lesson6.notebook November 27, 2012 Example: A ship steering 15 km/h [N70E]. If the current is 5 km/h [W10N], determine the relative velocity of the boat when standing on shore. Homework: pg 141, # 2 4, 6 10, 13 15, 18, 24, 25 pg 149, # 2 8, 13

24 Lesson7.notebook November 27, 2012 The Dot Product of Two Vectors Today's goal: I can recognize the need for the Dot Product and situations it is required. I can then apply it using previously discussed methods. Example: Determine the amount of work to move the object from point A to point B in the diagram where: f applied force AB displacement vector θ f B HINT: W = force x displacement Work is also a scalar! 20 A General Notes on the Dot Product (also known as the Scalar Product) For non zero vectors u and v, we define the Dot Product as: u v = u v cosθ, where θ is a value 0 θ 180. If u or v is the zero vector, than u v = 0. If θ=90, then cosθ = 0. If u and v are perpendicular to each other, the dot product is zero.

25 Lesson7.notebook November 27, 2012 Properties of the Dot Product 1) For non zero vectors u and v, if u v = 0, the vectors are perpendicular. 2) For any vectors u and v, u v = v u. The Dot Product is communative. 3) For any vector u, u u = u 2. 4) For any vectors u and v with a scalar value k, k εr, (ku) v = k(u v) = u (kv) 5) For any vectors u, v, and w, u (v + w) = u v + u w Applying the Dot Product If u = u x i + u y j + u z k and v = v x i + v y j + v z k then, Multiplying unit vectors: Performing the Dot Product: u v =

26 Lesson7.notebook November 27, 2012 Summary: If u = [a, b, c] and v = [d, e, f] then: u v = Example: Determine the Dot Product. 1) u = [3,7] and v = [4, 2] 2) u = 2i + 3j 4k and v = i + 6j + 2k

27 Lesson7.notebook November 27, 2012 Example: Determine the angle between the two vectors. 1) u = [2,4] and v = [ 7, 2] 2) u = [5, 2, 1] and v = [10, 3, 4] Example: Determine the value of z to make the vectors perpendicular. u = [2, 7, 6] and v = [5, 4, z] Homework: page 178 #1 15, 21

28 Lesson8.notebook November 27, 2012 The Cross Product of Two Vectors Today's goal: I can identify when the Cross Product would be required and how in principle it works. I can use the appropriate skills to determine its value. Example: A wrench is tightening a nut. Discuss the forces involved. Magnitude will depend on: 1) 2) Why is torque a vector? Resulting formula: The Right Hand Rule The Right Hand Rule is applied because we can model 3 vectors (ie. applied force, Torque or Moment, and lever length) using fingers from our hand. In general, we point our fingers along vector u (ie. the lever length) and then curl our fingers in the direction v (ie. applied force) for angles of 0 θ 180. The direction of our thumb then indicates the direction of the torque or moment (ie. is the bolt or screw tightening or loosening from the board).

29 Lesson8.notebook November 27, 2012 Notation u x v = u v sinθn, where: General Properties 1) u x v = (v x u) Not communative 2) For non zero vectors u and v, both vectors are collinear if, and only if, u x v = 0. 3) Two vectors are not collinear if u x v 0. 4) For vectors u, v, and w: u x (v + w) = (u + v) x w = ku x v = Applying the Cross Product For vectors u = [u x, u y, u z ] and v = [v x, v y, v z ]: u x v = = Easier forms to help remember:

30 Lesson8.notebook November 27, 2012 Example: Determine the Cross Product. 1) u = 5, v = 10, θ = 30 2) u = [3, 6, 2] and v = [ 2, 4, 8] 3) u = 2i 3j + 5k and v = 7i + 2j 3k Example: Determine a vector perpendicular to u = [ 1, 3, 2] and v = [ 3, 2, 1]. Justify your answer.

31 Lesson8.notebook November 27, 2012 Example: If u x v = 34, u = 4, and v = 9 the find θ. Example: Find a x b where a = 2i 3j + 5k and b = 7i + 2j 3k. Homework: page 185 #1 10, 14

32 Lesson9.notebook November 27, 2012 Applications of the Dot and Cross Product Today's goal: I can determine when the Dot and Cross Product need to be applied and understand how they are used in a variety of real world applications that are modelled in a simple format. The Dot and Cross product have a variety of applications where they can be applied individually or in combination. Determining 3 Space What is 3 space? Example: Determine if the given vectors are the basis for space: u = [2, 3, 5] v = [ 1, 0, 3] w = [ 4, 1, 0]

33 Lesson9.notebook November 27, 2012 Projections We talked about the x and y components of a vector also being the projection (or shadow) onto the x and y axis. What about the projection of one vector onto another? For example... Area of a Parallelogram With drafting programs, they often have the ability to calculate the area of a given shape. This is how they work... Area = w h b

34 Lesson9.notebook November 27, 2012 Volume of a Parallelepiped (or any other rectangular prism) Similar to area, programs can also calculate volume. This is how they work... Volume = Work Torque

35 Lesson9.notebook November 27, 2012 Example: If u = [ 1, 3, 6] and v = [ 2, 1, 3], find the projection of u onto v and the magnitude of the projection. Example: A parallelogram has vertices A(1, 1, 1), B(4, 3, 2), C(1, 2, 3) and D( 2, 0, 4). Determine the area.

36 Lesson9.notebook November 27, 2012 Example: Use the Dot Product to prove the cosine law. Example: Determine the Torque for the following.

37 Lesson9.notebook November 27, 2012 Homework: Proof questions: 1) Prove that: u x v 2 = u 2 v 2 (u v) 2 2) Prove that: a + b 2 + a b 2 = 2 a b 2 3) Prove that: a 2 b 2 (a b) 2 = 0 4) ABC is an isosceles triangle with base BC. Use vectors to prove that the median from vertex A is perpendicular to the base. page 192 #1 3, 7, 9 18 Unit Test: End date of unit: Test dates: Review dates: Review material: Handout pg 153 (pick and choose) pg 155 (pick and choose) pg 194 (pick and choose) pg 197 (pick and choose)