Vectors

1. Vectors Chapter 46 ch46 Vectors by Chtan FYKulai

2. A VECTOR? • Describes the motion of an object • A Vector comprises • Direction • Magnitude • We will consider • Column Vectors • General Vectors • Vector Geometry Size ch46 Vectors by Chtan FYKulai

3. NOTE! Label is in BOLD. When handwritten, draw a wavy line under the label i.e. a Column Vectors Vector a 2 up 4 RIGHT COLUMN Vector ch46 Vectors by Chtan FYKulai

4. b Column Vectors Vector b 2 up 3 LEFT COLUMN Vector? ch46 Vectors by Chtan FYKulai

5. n Column Vectors Vector u 2 down 4 LEFT COLUMN Vector? ch46 Vectors by Chtan FYKulai

6. b a d c Describe these vectors ch46 Vectors by Chtan FYKulai

7. Alternative labelling F B D E G C A H ch46 Vectors by Chtan FYKulai

8. k k k k General Vectors A Vector has BOTH a Length & a Direction All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k k can be in any position ch46 Vectors by Chtan FYKulai

9. k General Vectors Line CD is Parallel to AB B CD is TWICE length of AB D A 2k Line EF is Parallel to AB E EF is equal in length to AB C -k EF is opposite direction to AB F ch46 Vectors by Chtan FYKulai

10. k Write these Vectors in terms of k B D 2k F G ½k 1½k E C -2k A H ch46 Vectors by Chtan FYKulai

11. B k D A C Combining Column Vectors ch46 Vectors by Chtan FYKulai

12. C B A Simple combinations ch46 Vectors by Chtan FYKulai

13. Q P R a b O Vector Geometry Consider this parallelogram Opposite sides are Parallel OQ is known as the resultant of a and b ch46 Vectors by Chtan FYKulai

14. Resultant of Two Vectors • Is the same, no matter which route is followed • Use this to find vectors in geometrical figures ch46 Vectors by Chtan FYKulai

15. . S is the Midpoint of PQ. Work out the vector Q S P R a b O e.g.1 = a + ½b ch46 Vectors by Chtan FYKulai

16. . Q S S is the Midpoint of PQ. Work out the vector P R a b O Alternatively - ½b = b + a = ½b + a = a + ½b ch46 Vectors by Chtan FYKulai

17. C p M Find BC = + A q B BC BA AC AC= p, AB = q e.g.2 M is the Midpoint of BC = -q + p = p - q ch46 Vectors by Chtan FYKulai

18. C p M Find BM = ½BC A q B BM AC= p, AB = q e.g.3 M is the Midpoint of BC = ½(p – q) ch46 Vectors by Chtan FYKulai

19. C p M Find AM + ½BC = A q B AB AM AC= p, AB = q e.g.4 M is the Midpoint of BC = q + ½(p – q) = q +½p - ½q = ½q +½p = ½(q + p) = ½(p + q) ch46 Vectors by Chtan FYKulai

20. C p M Find AM + ½CB = A q B AM AC AC= p, AB = q Alternatively M is the Midpoint of BC = p + ½(q – p) = p +½q - ½p = ½p +½q = ½(p + q) ch46 Vectors by Chtan FYKulai

21. Distribution’s law : The scalar multiplication of a vector : ch46 Vectors by Chtan FYKulai

22. Other important facts : ch46 Vectors by Chtan FYKulai

23. A vector with the starting point from the origin point is called position vector. 位置向量 ch46 Vectors by Chtan FYKulai

24. Every vector can be expressed in terms of position vector. ch46 Vectors by Chtan FYKulai

25. e.g.5 Given that , and also Find the values of ch46 Vectors by Chtan FYKulai

26. e.g.6 Given that ,, and are parallel. Find the value of ch46 Vectors by Chtan FYKulai

27. e.g.7 =, , a point . Find the coordinates of then express point in terms of . ch46 Vectors by Chtan FYKulai

28. e.g.8 If , , find the coordinates of ch46 Vectors by Chtan FYKulai

29. e.g.9 Given that ,, and are parallel. Find the value of ch46 Vectors by Chtan FYKulai

30. Magnitude of a vector ch46 Vectors by Chtan FYKulai

31. Unit vector : ch46 Vectors by Chtan FYKulai

32. e.g.10 Find the magnitude of the vectors : (b) ch46 Vectors by Chtan FYKulai

33. e.g.11 Find the unit vectors in e.g. 10 : (b) ch46 Vectors by Chtan FYKulai

34. Ratio theorem A P B ch46 Vectors by Chtan FYKulai

35. e.g.12 M is the midpoint of AB, find in terms of . ch46 Vectors by Chtan FYKulai

36. e.g.13 3 P divides AB into 2:3. Find in terms of . 2 ch46 Vectors by Chtan FYKulai

37. Application of vector in plane geometry e.g.14 In the diagram, CB=4CN, NA=5NX, M is the midpoint of AB. A M X B C N (a) Express the following vectors in terms of u and v ; (i) (ii) ch46 Vectors by Chtan FYKulai

38. (b) Show that (c) Calculate the value of (i) (ii) ch46 Vectors by Chtan FYKulai

39. Soln: (a) (i) (ii) (b) ch46 Vectors by Chtan FYKulai

40. (c) (i) (ii) ch46 Vectors by Chtan FYKulai

41. e.g.15 A M and N are midpoints of AB, AC. Prove that N M C B ch46 Vectors by Chtan FYKulai

42. e.g.16 B In the diagram K divides AD into 1:l, and divides BC into 1:k . 2a 1 A 1 K 6a l k D O C 2b 6b Express position vector OK in 2 formats. Find the values of k and l. ch46 Vectors by Chtan FYKulai

43. More exercises on this topic : 高级数学高二下册 Pg 33 Ex10g ch46 Vectors by Chtan FYKulai

44. Scalar product of two vectors If a and b are two non-zero vectors, θ is the angle between the vectors. Then , ch46 Vectors by Chtan FYKulai

45. Scalar product of vectors satisfying : Commutative law : Associative law : Distributive law : ch46 Vectors by Chtan FYKulai

46. e.g.17 Find the scalar product of the following 2 vectors : ch46 Vectors by Chtan FYKulai

47. e.g.18 If , find the angle between them. If are perpendicular, find k. ch46 Vectors by Chtan FYKulai

48. Scalar product (special cases) 1. Two perpendicular vectors Unit vector for y-axis N.B. Unit vector for x-axis ch46 Vectors by Chtan FYKulai

49. 2. Two parallel vectors N.B. ch46 Vectors by Chtan FYKulai

50. e.g.19 Given , Find . Ans:[17/2] ch46 Vectors by Chtan FYKulai