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GCSE: Vectors. Dr J Frost (jfrost@tiffin.kingston.sch.uk). Last modified: 26 th January 2014. 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.
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GCSE: Vectors Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 26th January 2014
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.
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 ? ?
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
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)
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. ?
ζ Dr Frost GCSE – Vectors – Lesson 2 Objectives: Be able to further manipulate vectors, particular when considering portions of vectors.
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. ?
Example 2 ? ?
Quick fire ratio P A B z The vector Find the following vectors given the specified ratios: ? ? ? ? ? ?
TEST YOUR UNDERSTANDING Vote with your diaries! (use the front for blue) A B C D
Given that M is the midpoint of BC, determine AM. B 6a M A 4b C 3a + 2b 3a + b 2a + 3b a + b
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
Exercises Exercise 2 on sheet.
ζ Dr Frost GCSE – Vectors – Lesson 3
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
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
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?
GCSE Mark Scheme (3)
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!
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”. ? ?