2. 6.3 Vectors in the Plane Day 1 2015

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity of the ball. The velocity is a vector quantity since it has both a magnitude and a direction. Vectors are used to represent velocity, force, tension, and many other quantities. A vector is a quantity with both a magnitude and a direction. Magnitude and Direction

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 A quantity with magnitude and direction is represented by a directed line segment PQ with initial point P and terminal point Q. Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude. u v Directed Line Segment The vector v = PQ is the set of all directed line segments of length ||PQ|| which are parallel to PQ. P Q

5. Vector Representation by Directed Line Segments Let u be represented by the directed line segment from P = (0,0) to Q = (3,2), and let v be represented by the directed line segment from R = (1,2) to S = (4,4). Show that u = v. 1 2 3 4 4 3 2 1 P Q R S u v Using the distance formula, show that u and v have the same length. Show that their slopes are equal. Slopes of u and v are both

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Scalar multiplication is the product of a scalar, or real number, times a vector. For example, the scalar 3 times v results in the vector 3v, three times as long and in the same direction as v. v 3v3v v The product of - and v gives a vector half as long as and in the opposite direction to v. - v Scalar Multiplication

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Vector Addition To add vectors u and v: 1. Place the initial point of u at the terminal point of v. 2. Draw the vector with the same initial point as v and the same terminal point as u. u v v + u v u Vector Addition v u

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Vector Subtraction as adding the opposite. To subtract vectors u and v: u  v Add the opposite of v to u: u +(  v) 1. Place the initial point of  v at the terminal point of u. 2. Draw the vector u  v from the initial point of u to the terminal point of v. Vector Subtraction v u -v-v u u  v -v-v u

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Vector Subtraction To subtract vectors u and v: 1. Place the initial point of v at the initial point of u. 2. Draw the vector u  v from the terminal point of v to the terminal point of u. Vector Subtraction v u v u v u u  v

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (v 1, v 2 ). This is the component form of a vector v, written as. Standard Position If v is a vector with initial point P = (p 1, p 2 ) and terminal point Q = (q 1, q 2 ), then 1. The component form of v is v = q 1  p 1, q 2  p 2 2. The magnitude (or length) of v is ||v|| = x y (v 1, v 2 ) x y P (p 1, p 2 ) Q (q 1, q 2 )

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Standard Position Remember that to write a vector in component form: v = q 1  p 1, q 2  p 2 Use terminal point – initial point.

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Find the component form and magnitude of the vector v with initial point P = (3,  2) and terminal point Q = (  1, 1). Example: Magnitude The magnitude of v is ||v|| = = = 5.

13. You Try: Find the component form and length of the vector v that has initial point (4,-7) and terminal point (-1,5) Let P = (4, -7) = (p 1, p 2 ) and Q = (-1, 5) = (q 1, q 2 ). Then, the components of v = are given by Thus, v = The length of v is

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Equal Vectors Example: If u = PQ, v = RS, and w = TU with P = (1, 2), Q = (4, 3), R = (1, 1), S = (3, 2), T = (-1, -2), and U = (1, -1), determine which of u, v, and w are equal. (solution follows.) Calculate the component form for each vector: u = 4  1, 3  2 = 3, 1 v = 3  1, 2  1 = 2, 1 w = 1  (-1),  1  (-2) = 2, 1 Therefore v = w but v = u and w = u. // Two vectors u = u 1, u 2 and v = v 1, v 2 are equal if and only if u 1 = v 1 and u 2 = v 2.

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Operations on Vectors in the Coordinate Plane Let u =, v =, and let c be a scalar. 1. Scalar multiplication cu = 2. Addition u + v = 3. Subtraction u  v =

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 x y u + v x y Examples: Given vectors u = and v = Find -2u, u+v, and u-v Try these on your own. -2u = u + v = + = u  v =  = Examples: Operations on Vectors (4, 2) u (-8, -4) -2u x y (6, 7) (2, 5) (4, 2) v u (2, 5) (4, 2) v u (2, -3) u  v

17. Vector Operations Ex. Let v = and w =. Find the following vectors. a. 2v b. w – v -22 6 10 8 4 2 -2 -4 v 2v 1 2 3 4 4 3 2 1 5 w -v w - v

18. u = unit vector Ex: Find a unit vector in the direction of v = A unit vector is a vector whose magnitude = 1. In many vector applications it is useful to find a unit vector that has the same direction as a given vector v. To do this, divide v by its length to obtain:

19. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 You try: a) Find a unit vector in the direction of b) Find the magnitude of the unit vector you just found. 1

20. Homework 6.3 Day 1 Pg.417 1-35 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19

21. Vectors in the Plane Day 2 2015 Digital Lesson

22. Precalculus HWQ Answer this question at the top of your homework, then turn the homework in. Find the unit vector in the same direction as Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 The unit vectors and are called the standard unit vectors. i represents one unit of horizontal movement and j represents one unit of vertical movement. Any vector can be represented by what is called a linear combination of unit vectors. Example: Vector can be represented as a linear combination of unit vectors by rewriting it as v= 2i-6j. Standard unit vectors

24. Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2, -5) and terminal point (-1, 3). Write u as a linear combination of the standard unit vectors of i and j. -22 4 6 4 2 -2 -4 -6 -8 (2, -5) u (-1, 3) Solution -22 6 10 8 4 2 -2 -4 Graphically, it looks like… -3i 8j

25. Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (1, -7) and terminal point (-1, 2).Write u as a linear combination of the standard unit vectors i and j. Begin by writing the component form of the vector u.

26. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 Example: Find the vector v with the given magnitude and the same direction as u.

27. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 You try: Find the vector v with the given magnitude and the same direction as u.

28. Vector Operations Let u = -3i + 8j and let v = 2i - j. Find 2u - 3v. 2u - 3v = 2(-3i + 8j) - 3(2i - j) = -6i + 16j - 6i + 3j = -12i + 19 j

29. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 Direction Angles If u is a unit vector such that is the angle (measured counter-clockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and you have:

30. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 x y Finding Direction Angles The direction angle  of a vector v is the angle formed by the positive half of the x-axis and the ray along which v lies. x y v θ v θ x y v x y (x, y) Direction Angle If v = 3, 4, then tan  = and  = 53.13 . If v = x, y, then tan  =.

31. Find the magnitude and direction angle of the vector: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30

32. Find the magnitude and direction angle of the vector: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31

33. Find the direction angle of the vector: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32

34. Find the direction angle of the vector: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33

35. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 34 Writing Vectors using Direction Angles If u is a unit vector such that is the angle (measured counter-clockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and you have: To indicate a length other than 1, multiply the unit vector by a magnitude. The vector has a magnitude of 5 in the direction of 30 degrees.

36. Find the component form of v given its magnitude and the angle it makes with the positive x-axis: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35

37. Homework 6.3 Day 2 Pg.418 37-61 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 36

38. Find the direction angle and write the component form using it: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 37

39. Vectors in the Plane Day 3 2015 (use class notes handout)

40. HWQ Add vectors u and v. u= 2i-j v = -i+j Explain how you found the sum. u+v = 1i+0j = I Add two vectors by adding the horizontal components (x), and the vertical (y) components. The result is a new vector called the resultant of u and v. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 39

41. The sum of two or more vectors is called the resultant of the vectors. Find the resultant of u and v. Resultant Vectors (sum of vectors)

42. Applications Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 41 1.A ship leaving port sails for 75 miles in a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector. A ship leaving port sails for 75 miles at a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector.

43. Applications Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 42 1.A ship leaving port sails for 75 miles in a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector. Find the component form of the vector that represents the velocity of a plane descending at a speed of 100 mph. @ an angle 30 degrees below horizontal. 1.Find the component form of the vector that represents the velocity of a plane descending at a speed of 100 mph. @ an angle of 30 degrees below horizontal.

44. Applications Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 43 1.A ship leaving port sails for 75 miles in a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector. An airplane is traveling at a speed of 500 mph. with an air bearing of 330 degrees, at a fixed altitude, with negligible wind velocity. As the airplane reaches a certain point, it encounters a wind blowing with a velocity of 70 mph. in the direction Find : a. The resultant speed. b. The direction of the airplane..

45. Applications Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 44 1.A ship leaving port sails for 75 miles in a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector. A piling for a high-rise building is pushed by two bulldozers at exactly the same time. One bulldozer exerts a force of 1550 pounds in a westerly direction. The other bulldozer pushes the piling with a force of 3050 pounds in a northerly direction. What is the magnitude of the resultant force upon the piling? What is the direction of the resulting force upon the piling?

46. You Try Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 45 1.A ship leaving port sails for 75 miles in a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector. The Shanghai World Finance Center building in Shanghai, China, is 1508 feet tall. Suppose that a piling for building is pushed by two bulldozers at exactly the same time. One bulldozer exerts a force of 900 pounds in an easterly direction. The other bulldozer pushes the piling with a force of 2150 pounds in a northerly direction. What is the magnitude of the resultant force upon the piling? What is the direction of the resulting force upon the piling?

47. Homework 6.3 Day 3 Pg.418 63-65 odd, 69-75odd,81,82 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 46