1. Higher Unit 3 Vectors and Scalars 3D Vectors Properties of vectors Properties 3D Adding / Sub of vectors Section formula Multiplication by a Scalar Scalar Product Unit Vector Component Form Position Vector Angle between vectors Collinearity Perpendicular Section Formula Properties of Scalar Product Exam Type Questions www.mathsrevision.com

2. Vectors & Scalars A vector is a quantity with BOTH magnitude (length) and direction. Examples : Gravity Velocity Force

3. Vectors & Scalars A scalar is a quantity that has magnitude ONLY. Examples : Time Speed Mass

4. Vectors & Scalars A vector is named using the letters at the end of the directed line segment or using a lowercase bold / underlined letter This vector is named u u or u or u

5. Also known as column vector Vectors & Scalars A vector may also be represented in component form. w z

6. Magnitude of a Vector A vector’s magnitude (length) is represented by A vector’s magnitude is calculated using Pythagoras Theorem. u

7. Vectors & Scalars Calculate the magnitude of the vector. w

8. Vectors & Scalars Calculate the magnitude of the vector.

9. Equal Vectors Vectors are equal only if they both have the samemagnitude ( length ) and direction. Vectors are equal if they have equal components. For vectors

10. Equal Vectors By working out the components of each of the vectors determine which are equal. a a b c d g g e f h

11. Addition of Vectors Any two vectors can be added in this way Arrows must be nose to tail b b a a + b

12. Addition of Vectors Addition of vectors B A C

13. Addition of Vectors In general we have For vectors u and v

14. Zero Vector The zero vector

15. Negative Vector Negative vector For any vector u

16. Subtraction of Vectors Any two vectors can be subtracted in this way u Notice arrows nose to nose v v u - v

17. Subtraction of Vectors Subtraction of vectors Notice arrows nose to nose a b a - b

18. Subtraction of Vectors In general we have For vectors u and v

19. Multiplication by a Scalar Multiplication by a scalar ( a number) Hence if u = kv then u is parallel to v Conversely if u is parallel to vthen u= kv

20. Multiplication by a Scalar Multiplication by a scalar Write down a vector parallel to a b Write down a vector parallel to b a

21. Multiplication by a Scalar Show that the two vectors are parallel. If z = kw then z is parallel to w

22. Multiplication by a Scalar Alternative method. If w = kz then w is parallel to z

23. Unit Vectors For ANY vector v there exists a parallel vector u of magnitude 1 unit. This is called the unit vector.

24. v Unit Vectors u Find the components of the unit vector, u , parallel to vector v , if So the unit vector is u

25. A Position Vectors B A is the point (3,4) and B is the point (5,2). Write down the components of Answers the same !

26. A Position Vectors a B b 0

27. A Position Vectors a B b 0

28. Position Vectors If P and Q have coordinates (4,8) and (2,3) respectively, find the components of

29. Position Vectors P Graphically P (4,8) Q (2,3) p q - p Q q O

30. Collinearity Reminder from chapter 1 Points are said to be collinear if they lie on the same straight line. For vectors

31. Collinearity Prove that the points A(2,4), B(8,6) and C(11,7) are collinear.

32. Collinearity

33. Section Formula B 3 2 S b 1 s A a O

34. General Section Formula B m + n n P b m p A a O

35. General Section Formula Summarising we have B n If p is a position vector of the point P that divides AB in the ratio m : n then P m A

36. General Section Formula A and B have coordinates (-1,5) and (4,10) respectively. Find P if AP : PB is 3:2 B 2 P 3 A

37. 3D Coordinates In the real world points in space can be located using a 3D coordinate system. For example, air traffic controllers find the location a plane by its height and grid reference. z (x, y, z) y x O

38. 3D Coordinates Write down the coordinates for the 7 vertices y z (0, 1, 2) E (6, 1, 2) A (0, 0, 2) F 2 B (6, 0, 2) H D (6, 1, 0) (0,0, 0) G 1 x C 6 (6, 0, 0) What is the coordinates of the vertex H so that it makes a cuboid shape. O H(0, 1, 0 )

39. 3D Vectors 3D vectors are defined by 3 components. For example, the velocity of an aircraft taking off can be illustrated by the vector v. z (7, 3, 2) 2 v y 2 3 3 x O 7 7

40. 3D Vectors Any vector can be represented in terms of the i, j and k Where i, j and k are unit vectors in the x, y and z directions. z y k j x i O

41. 3D Vectors Any vector can be represented in terms of the i, j and k Where i, j and k are unit vectors in the x, y and z directions. z (7, 3, 2) v y v = ( 7i+ 3j + 2k ) 2 3 x O 7

42. 3D Vectors Good News All the rules for 2D vectors apply in the same way for 3D.

43. Magnitude of a Vector A vector’s magnitude (length) is represented by A 3D vector’s magnitude is calculated using Pythagoras Theorem twice. z v y 1 2 O x 3

44. Addition of Vectors Addition of vectors

45. Addition of Vectors In general we have For vectors u and v

46. Negative Vector Negative vector For any vector u

47. Subtraction of Vectors Subtraction of vectors

48. Subtraction of Vectors For vectors u and v

49. Multiplication by a Scalar Multiplication by a scalar ( a number) Hence if u = kv then u is parallel to v Conversely if u is parallel to vthen u= kv

50. Multiplication by a Scalar Show that the two vectors are parallel. If z = kw then z is parallel to w