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![a. 𝐴𝐵 = −𝒂 + 𝒃
b. 𝐴𝑋 = −
1
5
𝒂 +
1
5
𝒃
c. 𝑂𝑋 =
4
5
𝒂 +
1
5
𝒃
d. 𝐵𝑋 =
4
5
𝒂 −
4
5
𝒃
a. 𝐴𝑌 = −
1
3
𝒂 +
1
3
𝒃
b. 𝑂𝑌 =
2
3
𝒂 +
1
3
𝒃
c. 𝑌𝑂 = −
2
3
𝒂 −
1
3
𝒃
1
2
?
?
?
?
?
?
?
Exercise 2 (on provided sheet)
𝑋 is a point such
that 𝐴𝑋: 𝑋𝐵 = 1: 4
𝑌 is a point such
that 𝑌𝐵 = 2𝐴𝑌
3
[June 2009 2H Q23]
a) Find 𝐴𝐵 in terms of 𝒂 and 𝒃.
𝐴𝐵 = −𝒂 + 𝒃 𝑜𝑟 𝒃 − 𝒂
b) 𝑃 is on 𝐴𝐵 such that 𝐴𝑃: 𝑃𝐵 = 3: 2.
Show that 𝑂𝑃 =
1
5
2𝒂 + 3𝒃
?
?](https://image.slidesharecdn.com/gcse-vectors-240423184703-c114b076/75/GCSE-Vectors-pptx-notes-for-both-foundation-tier-and-higher-tier-13-2048.jpg)
![Exercise 2 (on provided sheet)
5
a. 𝑂𝑅 =
3
5
𝒂 +
2
5
𝒃
b. 𝐵𝑆 =
1
4
𝒂
c. 𝑂𝑆 =
1
4
𝒂 + 𝒃
d. 𝑅𝑆 = −
7
20
𝒂 +
3
5
𝒃
6
a. 𝐷𝐶 = −𝒛 + 𝒚
b. 𝐷𝑀 = −
1
2
𝒛 +
1
2
𝒚
c. 𝐴𝑀 = 𝐴𝐷 + 𝐷𝑀 =
1
2
𝒛 +
1
2
𝒚
d. 𝐵𝑀 = 𝐵𝐴 + 𝐴𝑀 = −𝒙 +
1
2
𝒛 +
1
2
𝒚
e. 𝐵𝑃 =
2
3
𝐵𝑀 = −
2
3
𝒙 +
1
3
𝒛 +
1
3
𝒚
f. 𝐴𝑃 = 𝐴𝐵 + 𝐵𝑃 =
1
3
𝒙 +
1
3
𝒛 +
1
3
𝒚
?
?
?
?
?
?
?
?
?
?
𝑂𝐴𝐶𝐵 is a parallelogram. 𝑅
is a point such that
𝐴𝑅: 𝑅𝐵 = 2: 3. 𝑆 is a point
such that 𝐵𝑆: 𝑆𝐶 = 1: 3.
𝑀 is the midpoint of
𝐶𝐷, 𝐵𝑃: 𝑃𝑀 = 2: 1
4 [Nov 2010 1H Q27] 𝑀 is the
midpoint of 𝑂𝑃.
a) Express 𝑂𝑀 in terms of 𝒂 and 𝒃.
𝑂𝑀 =
1
2
𝒂 + 𝒃 𝑜𝑟
1
2
𝒂 +
1
2
𝒃
b) Express 𝑇𝑀 in terms of 𝒂 and 𝒃
giving your answer in its simplest
form.
𝑇𝑀 = −𝒂 +
1
2
𝑂𝑀
= −𝒂 +
1
2
𝒂 + 𝒃
=
1
2
𝒃 −
1
2
𝒂
?
?](https://image.slidesharecdn.com/gcse-vectors-240423184703-c114b076/75/GCSE-Vectors-pptx-notes-for-both-foundation-tier-and-higher-tier-14-2048.jpg)
GCSE-Vectors.pptx notes for both foundation tier and higher tier
- 1. GCSE: Vectors Dr JFrost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 20th March 2017
- 2. 𝑥 𝑦 1 2 3 4 5 6 -1 -2 1 2 34 5 6 7 2,3 4 1 Coordinates represent a position. We write 𝑥, 𝑦 to indicate the 𝑥 position and 𝑦 position. Vectors represent a movement. We write 𝑥 𝑦 to indicate the change in 𝑥 and change in 𝑦. ? ? ? ? Note we put the numbers vertically. Do NOT write 4 1 ; there is no line and vectors are very different to fractions! You may have seen vectors briefly before if you’ve done transformations; we can use them to describe the movement of a shape in a translation. Coordinates vs Vectors
- 3. 𝒂 𝒃 𝒄 𝒅 𝒆 𝒇 𝒂 = 1 3 𝒃 = 5 1 𝒄= 2 −3 𝒅 = −4 2 𝒆 = −4 −4 𝒇 = −1 1 𝒈 = 0 −3 𝒈 ? ? ? ? ? ? ? Vectors to represent movements
- 4. You’re used tovariables representing numbers in maths. They can also represent vectors! 𝒂 𝒃 𝑋 𝑌 What can you say about how we use variables for vertices (points) vs variables for vectors? We use capital letters for vertices and lower case letters for vectors. There’s 3 ways in which can represent the vector from point 𝑋 to 𝑍: 1. 𝒂 (in bold) 2. 𝑎 (with an ‘underbar’) 3. 𝑋𝑍 𝑍 ? ? ? ? Writing Vectors
- 5. 𝒂 𝒃 𝑋 𝑌 𝑍 𝑋𝑍 = 𝒂= 2 5 𝑍𝑌 = 𝒃 = 5 3 𝑋𝑌 = 7 8 ? ? ? What do you notice about the numbers in 7 8 when compared to 2 5 and 5 3 ? We’ve simply added the 𝑥 values and 𝑦 values to describe the combined movement. i.e. 𝒂 + 𝒃 = 2 5 + 5 3 = 7 8 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 ? Bro Important Note: The point is that we can use any route to get from the start to finish, and the vector will always be the same. • Route 1: We go from 𝑋 to 𝑌 via 𝑍. 𝑋𝑌 = 2 5 + 5 3 = 7 8 • Route 2: Use the direct line from 𝑋 to 𝑌: 7 8 Adding Vectors
- 6. 𝒂 Scaling Vectors We can‘scale’ a vector by multiplying it by a normal number, aptly known as a scalar. If 𝒂 = 4 3 , then 2𝒂 = 2 4 3 = 8 6 What is the same about 𝒂 and 2𝒂 and what is different? 2𝒂 Bro Note: Note that vector letters are bold but scalars are not. Same: Same direction / Parallel Different: The length of the vector, known as the magnitude, is longer. ? ? ?
- 7. More on Adding/SubtractingVectors 𝐴 𝑋 𝒂 𝒃 2𝒄 If 𝑂𝐴 = 𝒂, 𝐴𝐵 = 𝒃 and 𝑋𝐵 = 2𝒄, then find the following in terms of 𝑎, 𝑏 and 𝑐: 𝑂𝐵 = 𝒂 + 𝒃 𝑂𝑌 = 𝒂 + 2𝒃 𝐴𝑋 = 𝒃 − 2𝒄 𝑋𝑂 = 2𝒄 − 𝒃 − 𝒂 𝑌𝑋 = −𝒃 − 2𝒄 Note: Since − 𝑥 𝑦 = −𝑥 −𝑦 , subtracting a vector goes in the opposite direction. ? ? ? 𝒃 𝐵 𝑌 𝑂 𝒃 𝒂 Bro Side Note: Since 𝑏 + 𝑎 would end up at the same finish point, we can see 𝒃 + 𝒂 = 𝒂 + 𝒃 (i.e. vector addition, like normal addition, is ‘commutative’) ? ?
- 8. a. 𝐵𝐴 =−𝑎 b. 𝐴𝐶 = 𝑎 + 𝑏 c. 𝐷𝐵 = 2𝑎 − 𝑏 d. 𝐴𝐷 = −𝑎 + 𝑏 a. 𝑀𝐾 = −𝑎 − 𝑏 b. 𝑁𝐿 = 3𝑎 − 𝑏 c. 𝑁𝐾 = 2𝑎 − 𝑏 d. 𝐾𝑁 = −2𝑎 + 𝑏 a. 𝑍𝑋 = 𝑎 + 𝑏 b. 𝑌𝑊 = 𝑎 − 2𝑏 c. 𝑋𝑌 = −𝑎 + 𝑏 d. 𝑋𝑍 = −𝑎 − 𝑏 a. 𝐴𝐵 = −𝑎 + 𝑏 b. 𝐹𝑂 = −𝑎 + 𝑏 c. 𝐴𝑂 = −2𝑎 + 𝑏 d. 𝐹𝐷 = −3𝑎 + 2𝑏 2 3 4 5 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Exercise 1 (on provided sheet) 1 State the value of each vector. 𝒂 𝒃 𝒄 𝒅 𝒆 𝒇 𝒂 = 3 1 𝒃 = 3 −2 𝒄 = −3 0 𝒅 = 0 5 𝒆 = −1 4 𝒇 = −3 −4 ? ? ? ? ? ?
- 9. Starter Expand and simplifythe following expressions: 2 𝒂 + 𝒃 + 𝒃 = 2𝒂 + 3𝒃 1 2 𝒂 − 𝒃 + 𝒃 = 1 2 𝒂 + 1 2 𝒃 𝒂 + 1 2 𝒂 + 𝒃 = 3 2 𝒂 + 1 2 𝒃 𝒂 − 2 𝒂 − 2𝒃 = −𝒂 + 4𝒃 1 3 𝒂 + 2𝒃 − 𝒃 = 1 3 𝒂 − 1 3 𝒃 1 2 3 4 5 ? ? ? ? ?
- 10. The ‘Two Parter’exam question Many exams questions follow a two-part format: a) Find a relatively easy vector using skills from Lesson 1. b) Find a harder vector that uses a fraction of your vector from part (a). Edexcel June 2013 1H Q27 𝑆𝑄 = −𝒃 + 𝒂 For (b), there’s two possible paths to get from 𝑁 to 𝑅: via 𝑆 or via 𝑄. But which is best? In (a) we found 𝑺 to 𝑸 rather than 𝑸 to 𝑺, so it makes sense to go in this direction so that we can use our result in (a). a b 𝑁𝑅 = 2 5 𝑆𝑄 + 𝒃 = 2 5 −𝒃 + 𝒂 + 𝒃 = 2 5 𝒂 + 3 5 𝒃 Bro Tip: This ratio wasn’t in the original diagram. I like to add the ratio as a visual aid. ? ? Bro Workings Tip: While you’re welcome to start your working with the second line, I recommend the first line so that your chosen route is clearer. 𝑄𝑅 is also 𝑏 because it is exactly the same movement as 𝑃𝑆. ?
- 11. Test Your Understanding 𝐴𝐵= 𝐴𝑂 + 𝑂𝐵 = −𝒂 + 𝒃 ? 𝑂𝑃 = 𝒂 + 3 4 𝐴𝐵 = 𝒂 + 3 4 −𝒂 + 𝒃 = 1 4 𝒂 + 3 4 𝒃 ? You MUST expand and simplify. Edexcel June 2012
- 12. Further Test YourUnderstanding 𝒂 𝒃 𝑂 𝐴 𝐵 𝑋 𝑂𝑋: 𝑋𝐵 = 1: 3 𝐴𝑋 = −𝒂 + 1 4 𝑂𝐵 = −𝒂 + 1 4 𝒂 + 𝒃 = −𝒂 + 1 4 𝒂 + 1 4 𝒃 = − 3 4 𝒂 + 1 4 𝒃 First Step? ? 𝒂 𝒃 𝑂 𝐴 𝐵 𝑌 𝐴𝑌: 𝑌𝐵 = 2: 3 𝑂𝑌 = 𝒂 + 2 5 𝐴𝐵 = 𝒂 + 2 5 −𝒂 + 𝒃 = 𝒂 − 2 5 𝒂 + 2 5 𝒃 = 3 5 𝒂 + 2 5 𝒃 First Step? ? A B
- 13. a. 𝐴𝐵 =−𝒂 + 𝒃 b. 𝐴𝑋 = − 1 5 𝒂 + 1 5 𝒃 c. 𝑂𝑋 = 4 5 𝒂 + 1 5 𝒃 d. 𝐵𝑋 = 4 5 𝒂 − 4 5 𝒃 a. 𝐴𝑌 = − 1 3 𝒂 + 1 3 𝒃 b. 𝑂𝑌 = 2 3 𝒂 + 1 3 𝒃 c. 𝑌𝑂 = − 2 3 𝒂 − 1 3 𝒃 1 2 ? ? ? ? ? ? ? Exercise 2 (on provided sheet) 𝑋 is a point such that 𝐴𝑋: 𝑋𝐵 = 1: 4 𝑌 is a point such that 𝑌𝐵 = 2𝐴𝑌 3 [June 2009 2H Q23] a) Find 𝐴𝐵 in terms of 𝒂 and 𝒃. 𝐴𝐵 = −𝒂 + 𝒃 𝑜𝑟 𝒃 − 𝒂 b) 𝑃 is on 𝐴𝐵 such that 𝐴𝑃: 𝑃𝐵 = 3: 2. Show that 𝑂𝑃 = 1 5 2𝒂 + 3𝒃 ? ?
- 14. Exercise 2 (onprovided sheet) 5 a. 𝑂𝑅 = 3 5 𝒂 + 2 5 𝒃 b. 𝐵𝑆 = 1 4 𝒂 c. 𝑂𝑆 = 1 4 𝒂 + 𝒃 d. 𝑅𝑆 = − 7 20 𝒂 + 3 5 𝒃 6 a. 𝐷𝐶 = −𝒛 + 𝒚 b. 𝐷𝑀 = − 1 2 𝒛 + 1 2 𝒚 c. 𝐴𝑀 = 𝐴𝐷 + 𝐷𝑀 = 1 2 𝒛 + 1 2 𝒚 d. 𝐵𝑀 = 𝐵𝐴 + 𝐴𝑀 = −𝒙 + 1 2 𝒛 + 1 2 𝒚 e. 𝐵𝑃 = 2 3 𝐵𝑀 = − 2 3 𝒙 + 1 3 𝒛 + 1 3 𝒚 f. 𝐴𝑃 = 𝐴𝐵 + 𝐵𝑃 = 1 3 𝒙 + 1 3 𝒛 + 1 3 𝒚 ? ? ? ? ? ? ? ? ? ? 𝑂𝐴𝐶𝐵 is a parallelogram. 𝑅 is a point such that 𝐴𝑅: 𝑅𝐵 = 2: 3. 𝑆 is a point such that 𝐵𝑆: 𝑆𝐶 = 1: 3. 𝑀 is the midpoint of 𝐶𝐷, 𝐵𝑃: 𝑃𝑀 = 2: 1 4 [Nov 2010 1H Q27] 𝑀 is the midpoint of 𝑂𝑃. a) Express 𝑂𝑀 in terms of 𝒂 and 𝒃. 𝑂𝑀 = 1 2 𝒂 + 𝒃 𝑜𝑟 1 2 𝒂 + 1 2 𝒃 b) Express 𝑇𝑀 in terms of 𝒂 and 𝒃 giving your answer in its simplest form. 𝑇𝑀 = −𝒂 + 1 2 𝑂𝑀 = −𝒂 + 1 2 𝒂 + 𝒃 = 1 2 𝒃 − 1 2 𝒂 ? ?
- 15. a. 𝐴𝐵 =−𝒂 + 𝒃 b. 𝐵𝐶 = 𝒂 c. 𝑀𝐵 = − 1 2 𝒂 + 1 2 𝒃 d. 𝑀𝑋 = − 1 6 𝒂 + 1 2 𝒃 e. 𝑋𝐴 = 2 3 𝒂 − 𝒃 f. 𝐶𝑀 = − 1 2 𝒂 − 1 2 𝒃 g. 𝑋𝑂 = − 1 3 𝒂 − 𝒃 7 ? 8 9 ? ? ? ? ? ? a. 𝑂𝐵 = 𝒂 + 2𝒃 b. 𝐵𝐶 = 𝒂 − 𝒃 c. 𝐴𝑀 = 1 2 𝒂 + 𝒃 d. 𝑂𝑀 = 3 2 𝒂 + 𝒃 ? ? ? ? a. 𝑂𝐶 = −𝒂 + 𝒃 b. 𝑋𝐶 = − 3 4 𝒂 + 1 2 𝒃 c. 𝐴𝑋 = − 5 4 𝒂 + 1 2 𝒃 ? ? ? Exercise 2 (on provided sheet)
- 16.
- 17. Types of vector‘proof’ questions “Prove that … is a straight line.” “Show that … is parallel to …” “Prove these two vectors are equal.”
- 18. ? Showing vectors areequal There is nothing mysterious about this kind of ‘prove’ question. Just find any vectors involved, in this case, 𝑂𝑋 and 2 5 𝑂𝑌. Hopefully they should be equal! Edexcel March 2013 2 5 𝑂𝑌 = 2 5 6𝑏 + 5𝑎 − 𝑏 = 2𝑎 + 2𝑏 𝑂𝑋 = 3𝑎 + 1 3 −3𝑎 + 6𝑏 = 2𝑎 + 2𝑏 ∴ 𝑂𝑋 = 2 5 𝑂𝑌 With proof questions you should restate the thing you are trying to prove, as a ‘conclusion’. ?
- 19. What do younotice? 𝒂 2𝒂 𝒂 + 2𝒃 2𝒂 + 4𝒃 𝒂 − 3 2 𝒂 ! Vectors are parallel if one is a multiple of another. ?
- 20. Parallel or notparallel? Vector 1 Vector 2 Parallel? (in general) 𝑎 + 𝑏 𝑎 + 2𝑏 𝑎 + 𝑏 3𝑎 + 3𝑏 𝑎 + 2𝑏 −3𝑎 − 6𝑏 𝑎 − 𝑏 −𝑎 + 𝑏 No Yes No Yes No Yes No Yes For ones which are parallel, show it diagrammatically.
- 21. How to showtwo vectors are parallel 𝒂 𝑴 𝑶 𝑩 𝑿 𝒃 𝑨 𝑪 𝑋 is a point on 𝐴𝐵 such that 𝐴𝑋: 𝑋𝐵 = 3: 1. 𝑀 is the midpoint of 𝐵𝐶. Show that 𝑋𝑀 is parallel to 𝑂𝐶. 𝑂𝐶 = 𝑎 + 𝑏 𝑋𝑀 = 1 4 −𝑎 + 𝑏 + 1 2 𝑎 = 1 4 𝑎 + 1 4 𝑏 = 1 4 𝑎 + 𝑏 𝑋𝑀 is a multiple of 𝑂𝐶 ∴ parallel. The key is to factor out a scalar such that we see the same vector. For any proof question always find the vectors involved first, in this case 𝑋𝑀 and 𝑂𝐶. The magic words here are “is a multiple of”. ?
- 22. Test Your Understanding 𝑃 𝐵 𝑂 𝐴 2𝒂 3𝒃 a)Find 𝐴𝐵 in terms of 𝒂 and 𝒃. −2𝒂 + 3𝒃 b) 𝑃 is the point on 𝐴𝐵 such that 𝐴𝑃: 𝑃𝐵 = 2: 3. Show that 𝑂𝑃 is parallel to the vector 𝒂 + 𝒃. ? Edexcel June 2011 Q26 Bro Exam Note: Notice that the mark scheme didn’t specifically require “is a multiple of” here (but write it anyway!), but DID explicitly factorise out the 6 5 ?
- 23. Proving three pointsform a straight line Points A, B and C form a straight line if: 𝑨𝑩 and 𝑩𝑪 are parallel (and B is a common point). Alternatively, we could show 𝑨𝑩 and 𝑨𝑪 are parallel. This tends to be easier. A B C ? A B C
- 24. 𝐴𝑁 = 2𝒃,𝑁𝑃 = 𝒃 𝐵 is the midpoint of 𝐴𝐶. 𝑀 is the midpoint of 𝑃𝐵. a) Find 𝑃𝐵 in terms of 𝒂 and 𝒃. b) Show that 𝑁𝑀𝐶 is a straight line. Straight Line Example −3𝒃 + 𝒂 𝑁𝑀 = 𝑏 + 1 2 𝑎 − 3𝑏 = 𝑏 + 1 2 𝑎 − 3 2 𝑏 = 1 2 𝑎 − 1 2 𝑏 = 𝟏 𝟐 (𝒂 − 𝒃) 𝑁𝐶 = −2𝑏 + 2𝑎 = 𝟐 𝒂 − 𝒃 𝑵𝑪 is a multiple of 𝑵𝑴 ∴ 𝑵𝑪 is parallel to 𝑵𝑴. 𝑵 is a common point. ∴ 𝑵𝑴𝑪 is a straight line. b a ? ?
- 25. November 2013 1HQ24 𝑂𝐴 = 𝑎 and 𝑂𝐵 = 𝑏 𝐷 is the point such that 𝐴𝐶 = 𝐶𝐷 The point 𝑁 divides 𝐴𝐵 in the ratio 2: 1. (a) Write an expression for 𝑂𝑁 in terms of 𝒂 and 𝒃. 𝑶𝑵 = 𝒃 + 𝟏 𝟑 −𝒃 + 𝒂 = 𝟏 𝟑 𝒂 + 𝟐 𝟑 𝒃 (b) Prove that 𝑂𝑁𝐷 is a straight line. 𝑶𝑫 = 𝒂 + 𝟐𝒃 𝑶𝑵 = 𝟏 𝟑 (𝒂 + 𝟐𝒃) 𝑶𝑵 is a multiple of 𝑶𝑫 and 𝑶 is a common point. ∴ 𝑶𝑵𝑫 is a straight line. ? ? Test Your Understanding
- 26. Exercise 3 𝑇 isthe point on 𝐴𝐵 such that 𝐴𝑇: 𝑇𝐵 = 5: 1. Show that 𝑂𝑇 is parallel to the vector 𝒂 + 2𝒃. 𝑂𝑇 = 2𝒃 + 1 6 −2𝒃 + 5𝒂 = 2𝒃 − 1 3 𝒃 + 5 6 𝒂 = 5 6 𝒂 + 5 3 𝒃 = 5 6 𝒂 + 2𝒃 1 𝑀 is the midpoint of 𝑂𝐴. 𝑁 is the midpoint of 𝑂𝐵. Prove that 𝐴𝐵 is parallel to 𝑀𝑁. 𝐴𝐵 = −2𝒎 + 2𝒏 = 2 −𝒎 + 𝒏 𝑀𝑁 = −𝒎 + 𝒏 𝐴𝐵 is a multiple of 𝑀𝑁 ∴ parallel. 2 ? ?
- 27. Exercise 3 3 𝐴𝐶𝐸𝐹 isa parallelogram. 𝐵 is the midpoint of 𝐴𝐶. 𝑀 is the midpoint of 𝐵𝐸. Show that 𝐴𝑀𝐷 is a straight line. 𝐷𝑀 = −𝒃 + 1 2 3𝒃 + 𝒂 = −𝒃 + 3 2 𝒃 + 1 2 𝒂 = 1 2 𝒂 + 1 2 𝒃 = 1 2 𝒂 + 𝒃 𝐷𝐴 = 2𝒃 + 2𝒂 = 2 𝒂 + 𝒃 𝐷𝐴 is a multiple of 𝐷𝑀 and 𝐷 is a common point, so 𝐴𝑀𝐷 is a straight line. 4 𝐶𝐷 = 𝒂, 𝐷𝐸 = 𝒃 and 𝐹𝐶 = 𝒂 − 𝒃 i) Express 𝐶𝐸 in terms of 𝒂 and 𝒃. 𝒂 + 𝒃 ii) Prove that 𝐹𝐸 is parallel to 𝐶𝐷. 𝐹𝐸 = 𝑎 − 𝑏 + 𝑎 + 𝑏 = 2𝒂 which is a multiple of 𝐶𝐷 iii) 𝑋 is the point on 𝐹𝑀 such that such that 𝐹𝑋: 𝑋𝑀 = 4: 1. Prove that 𝐶, 𝑋 and 𝐸 lie on the same straight line. 𝐶𝑋 = −𝒂 + 𝒃 + 4 5 2𝒂 − 1 2 𝒃 = 3 5 𝒂 + 3 5 𝒃 = 3 5 𝒂 + 𝒃 𝐶𝐸 = 𝒂 + 𝒃 ? ? ? ?
- 28. Exercise 3 5 𝑂𝐴𝐵𝐶 isa parallelogram. 𝑃 is the point on 𝐴𝐶 such that 𝐴𝑃 = 2 3 𝐴𝐶. i) Find the vector 𝑂𝑃. Give your answer in terms of 𝒂 and 𝒄. 𝑂𝑃 = 6𝒂 + 2 3 −6𝒂 + 6𝒄 = 2𝒂 + 4𝒄 = 2 𝒂 + 2𝒄 ii) Given that the midpoint of 𝐶𝐵 is 𝑀, prove that 𝑂𝑃𝑀 is a straight line. 𝑂𝑀 = 3𝒂 + 6𝒄 = 3 𝒂 + 2𝒄 𝑂𝑀 is a multiple of 𝑂𝑃 and 𝑂 is a common point, therefore 𝑂𝑃𝑀 is a straight line. 6 𝑂𝐴 = 3𝒂 and 𝐴𝑄 = 𝒂 and 𝑂𝐵 = 𝒃 and 𝐵𝐶 = 1 2 𝒃. 𝑀is the midpoint of 𝑄𝐵. Prove that 𝐴𝑀𝐶 is a straight line. 𝐴𝑀 = 𝒂 + 1 2 −4𝒂 + 𝒃 = −𝒂 + 1 2 𝒃 𝐴𝐶 = −3𝒂 + 3 2 𝒃 = 3 −𝒂 + 1 2 𝒃 𝐴𝐶 is a multiple of 𝐴𝑀 and 𝐴 is a common point, therefore 𝐴𝑀𝐶 is a straight line. ? ? ?