Chapter 12 Vectors

1. Chapter 12Vectors A. Vectors and scalars B. Geometric operations with vectors C. Vectors in the plane D. The magnitude of a vector E. Operations with plane vectors F. The vector between two points G. Vectors in space H. Operations with vectors in space I. Parallelism J. The scalar product of two vectors

2. An airplane in calm conditions is flying at 800 km/hr due east. A cold wind suddenly blows from the south-west at 35 km/hr, pushing the airplane slightly off course. Things to think about: a. How can we illustrate the plane’s movement and the wind using a scale diagram? b. What operation do we need to perform to find the effect of the wind on the airplane? c. Can you use a scale diagram to determine the resulting speed and direction of the airplane? Opening Problem

3. Quantities which have only magnitude are called scalars. Quantities which have both magnitude and direction are called vectors. Vectors and Scalars

4. The speed of the plane is a scalar. It describes its size or strength. The velocity of the plane is a vector. It includes both its speed and also its direction. Other examples of vector quantities are: acceleration force Displacementmomentum

5. We can represent a vector quantity using a directed line segment or arrow. The length of the arrow represents the size or magnitude of the quantity, and the arrowhead shows its direction. Directed Line Segment Representation

6. For example, farmer Giles needs to remove a fence post. He starts by pushing on the post sideways to loosen the ground. Giles has a choice of how hard to push the post and in which direction. The force he applies is therefore a vector. If farmer Giles pushes the post with a force of 50 Newtons (N) to the north-east, we can draw a scale diagram of the force relative to the north line. N 50N 45o Scale: 1 cm represents 25 N

7. Vector Notation

8. Two vectors are equal if they have the same magnitude and direction. Equal vectors are parallel and in the same direction, and are equal in length. The arrows that represent them are translations of one another. Geometric vector equality

9. AB and BA have the same length, but they have opposite directions. B B A A Geometric negative vectors

10. A typical problem could be: A runner runs east for 4 km and then south for 2 km. How far is she from her starting point and in what direction? 4 km N W E 2 km Geometric operations with vectors x km S

11. Suppose we have three towns P, Q, and R. A trip from P to Q followed by a trip from Q to R has the same origin and destination as a trip from P to R. This can be expressed in vector form as the sum PQ + QR = PR. P R Q Geometric vector addition

12. To construct a + b: Step 1: Draw a. Step 2: At the arrowhead end of a, draw b. Step 3: Join the beginning of a to the arrowhead end of b. This is vector a + b. “head-to-tail” method of vector addition

13. Given a and b as shown, construct a + b. a b

14. b a b a a + b

15. The zero vector 0 is a vector of length 0. For any vector a: a + 0 = 0 + a = a a +(-a) =(-a) + a = 0. THE ZERO VECTOR

16. Find a single vector which is equal to: • BC + CA • BA + AE + EC • AB + BC + CA • AB + BC + CD + DE E D C A B

17. BC +CA = BA E D C A B

18. E D BA + AE + EC = BC C AB + BC + CA = AA = 0 AB + BC + CD + DE = AE A B

19. To subtract one vector from another, we simply add its negative. a – b = a +(-b) b -b a a b a - b Geometric vector subtraction

20. For r, s, and t shown, find geometrically: a. r – s b. s – t – r r t s

21. r – s s r -s r – s

22. s – t – r r -t s – t – r t s

23. For points A, B, C, and D, simplify the following vector expressions: a. AB – CB b. AC – BC – DB

24. a. Since –CB = BC, then AB – CB = AB + BC = AC. b. Same argument as part a. AC – BC – DB = AC + CB + BD = AD

25. Whenever we have vectors which form a closed polygon, we can write a vector equation which relates the variables. Vector Equation

26. Find, in terms of r, s, and t: a. RS b. SR c. ST S R r s O T t

27. Start at R, go to O by –r then go to S by s. Therefore RS = -r + s = s – r. • Start at S, go to O by –s then go to R by r. Therefore SR = -s + r = r – s. • Start at S, go to O by –s then go to T by t. Therefore ST = -s + t = t – s.

28. If a is a vector and k is a scalar, then ka is also a vector and we are performing scalar multiplication. If k > 0, ka and a have the same direction. If k < 0, ka and a have opposite directions. If k = 0, ka = 0, the zero vector. Geometric scalar multiplication

29. Given vectors r and s, construct geometrically: a. 2r + s b. r – 3s s r

30. a. 2r + s s r s 2r 2r + s

31. b. r – 3s s -3s r – 3s r r

32. In transformation geometry, translating a point a units in the x-direction and b units in the y-direction can be achieved using the translation vector Vectors in the plane

33. Base unit vector All vectors in the plane can be described in terms of the base unit vectors i and j.

35. a. 7i + 3j 3j 7i

36. b. -6i c. 2i – 5j d. 6j e. -6i + 3j f. -5i – 5j

37. v v2 The magnitude of a vector v1

41. A unit vector is any vector which has a length of one unit. are the base unit vectors in the positive x and y-directions respectively. Unit vectors

42. Find k given that is a unit vector.

43. Knowing that is a unit vector, then

44. a+b b a2+b2 b2 a Operations with plane vectors b1 a2 a1 a1+b1

45. Check your answer graphically.

47. Algebraic negative vectors

48. Algebraic vector subtraction