GCSE: Vectors

1. GCSE: Vectors Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 26th January 2014

2. Starter b ( ) ( ) ( ) ( ) ( ) ( ) -1 1 -4 -4 -4 2 5 1 2 -3 1 3 ? ? c = a = d = e = f = b = a d ? ? f e ? ? c • Bro Tip: Ensure you can distinguish between coordinates and vectors. • Coordinates represent positions. • Vectors represent movement.

3. What is a vector? A vector is an entity with 2 properties: Direction Magnitude (the length) ? ? Vectors are equal if: Same direction & magnitude Vectors are parallel if: Same/opposite direction ? ?

4. Writing Vectors Just like conventional algebra, we can represent vectors as variables. Y There’s 3 ways in which can represent the vector from point X to Z: (in bold) (with an ‘underbar’) b Z a X

5. Adding/Subtracting/Scaling Vectors Determine: XY = a + b ? XZ = 2a ? XR = 2a + 2b Click to Start Bromanimation T R XQ = 2a + b ? M 3 2 XM = a + b ? Y Q YZ = a – b ? b b b b b b RS = -2b – a ? a a a a a a 1 2 X S Z ? MQ = - b+a (M is the midpoint of the line YT)

6. Parallel or not parallel? We earlier said that vectors are parallel if they have the same direction (but could have different magnitudes).  No  Yes  No  Yes No  Yes   No  Yes For ones which are parallel, show it diagrammatically. Therefore parallel (in the context of scalars) if: we can multiply one vector by a scalar to get the other. ?

7. Exercises (on your sheet)

8. ζ Dr Frost GCSE – Vectors – Lesson 2 Objectives: Be able to further manipulate vectors, particular when considering portions of vectors.

9. Example 1 is the midpoint of and the midpoint of . and Determine: ? ? ? ? ? Bro Tip: For more complicated vectors, express it as a sum of other vectors. ?

10. Example 2 ? ?

11. Example 3 ?

12. Quick fire ratio P A B z The vector Find the following vectors given the specified ratios: ? ? ? ? ? ?

13. TEST YOUR UNDERSTANDING Vote with your diaries! (use the front for blue) A B C D

14. Given that M is the midpoint of BC, determine AM. B 6a M A 4b C 3a + 2b 3a + b 2a + 3b a + b

15. Given that M is the midpoint of BC and Q is the point such that AQ:QB = 3:1, determine MQ. B Q 6a M A 4b C 1.5a – 3b a – 2b 1.5a – 2b a – 3b

16. Exercises Exercise 2 on sheet.

17. ζ Dr Frost GCSE – Vectors – Lesson 3

18. A challenging one! The ratio of the lengths OM to MQ is 3:2. The ratio of the lengths PN to NR is 4:1. Find MN P a N R O b c ? M Q

19. Recap Two vectors are parallel if: One is a multiple of the other. ? Points A, B and C form a straight line if: and are parallel (and B is a common point). ? C B A

20. More complicated question

21. GCSE Mark Scheme 1 mark 1 mark 1 mark Simplifying them to and Write this down! 1 mark “NM is a multiple of MC” (+ they have a common point M) Why do you think this is significant?

22. GCSE Mark Scheme (3)

23. GCSE Mark Scheme 1 mark 2a + 0.4(3b – 2a) or 3b + 0.6(2a – 3b) 1 mark Simplified to 1.2a + 1.2b 1 mark Explicitly stating that “1.2(a + b) is parallel to a + b” You need to explicitly state the conclusion!

24. Test your understanding C B A The approach to prove that ABC is a straight line is to: Prove that AB is parallel to BC. We can show that these are parallel by stating: “Vector BC is a multiple of AB”. ? ?