1. Vectors Vectors and Scalars Properties of vectors Adding / Sub of vectors Multiplication by a Scalar Position Vector Collinearity Section Formula

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

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

4. 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 or u oru u u

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

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 same magnitude ( 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 b c d e f g h a g

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

12. Addition of Vectors Addition of vectors A B 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 v u - v Notice arrows nose to nose v

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

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 v then u = kv

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

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

24. Position Vectors B A a b 0

25. B A a b 0

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

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

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

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

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

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

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

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

35. Addition of Vectors Addition of vectors

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

37. Negative Vector Negative vector For any vector u

38. Subtraction of Vectors Subtraction of vectors

39. Subtraction of Vectors For vectors u and v

40. 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 v then u = kv

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