1. Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com
GCSE: Vectors
Dr J Frost
Last modified: 20th March 2017

2. Coordinates vs Vectors𝑦 6 ?
Coordinates represent a position. We write 𝑥,𝑦 to indicate the 𝑥 position and 𝑦 position. 5 4 ? ?
2,3
Vectors represent a movement. We write 𝑥 𝑦 to indicate the change in 𝑥 and change in 𝑦. 3 ?
4 1 2
Note we put the numbers vertically. Do NOT write ; 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. 1 𝑥 -1 -2

3. Vectors to represent movements𝒃 𝒂
𝒂= 𝒃= 𝒄= 2 − 𝒅= −4 2 𝒆= −4 − 𝒇= −1 1 𝒈= 0 −3 ? ? 𝒅 ? ? 𝒇 ? ? 𝒈 𝒆 ? 𝒄

4. Writing Vectors
You’re used to variables representing numbers in maths. They can also represent vectors! 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:
𝒂 (in bold)
𝑎 (with an ‘underbar’) 𝑋𝑍 𝒃 𝑍 ? 𝒂 ? ? 𝑋 ?

5. Adding Vectors ? ? 𝑌 ? 𝑍 ? 𝑋 𝒃 𝒂 𝑋𝑍 =𝒂= 2 5 𝑍𝑌 =𝒃= 5 3 𝑋𝑌 = 7 8
𝑋𝑍 =𝒂= 𝑍𝑌 =𝒃= 𝑋𝑌 = 7 8 ? ? 𝑌 ? 𝒃
What do you notice about the numbers in when compared to and ?
We’ve simply added the 𝑥 values and 𝑦 values to describe the combined movement.
i.e.
𝒂+𝒃= = 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 𝑍 ? 𝒂
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. 𝑋𝑍 = = 7 8
Route 2: Use the direct line from 𝑋 to 𝑌: 7 8 𝑋

6. Scaling Vectors
We can ‘scale’ a vector by multiplying it by a normal number, aptly known as a scalar.
If 𝒂= , then
2𝒂= = 8 6
What is the same about 𝒂 and 2𝒂 and what is different? ? 𝒂 2𝒂
Same:
Same direction / Parallel
Different:
The length of the vector, known as the magnitude, is longer. ? ?
Bro Note: Note that vector letters are bold but scalars are not.

7. More on Adding/Subtracting Vectors𝑋
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 2𝒄 𝐴 𝐵 𝒃 𝒃 𝑌 ? 𝒂 ? 𝒂 ? 𝑂 ? 𝒃 ?
Note: Since − 𝑥 𝑦 = −𝑥 −𝑦 , subtracting a vector goes in the opposite direction.
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. Exercise 1
(on provided sheet) 1
State the value of each vector. 2 4 𝒂 𝒃
𝐵𝐴 =−𝑎
𝐴𝐶 =𝑎+𝑏
𝐷𝐵 =2𝑎−𝑏
𝐴𝐷 =−𝑎+𝑏 ?
𝑍𝑋 =𝑎+𝑏
𝑌𝑊 =𝑎−2𝑏
𝑋𝑌 = −𝑎+𝑏
𝑋𝑍 =−𝑎−𝑏 ? 𝒄 ? ? ? ? ? ? 𝒅 3 5 𝒆 𝒇
𝑀𝐾 =−𝑎−𝑏
𝑁𝐿 =3𝑎−𝑏
𝑁𝐾 =2𝑎−𝑏
𝐾𝑁 =−2𝑎+𝑏 ? ? ?
𝐴𝐵 =−𝑎+𝑏
𝐹𝑂 =−𝑎+𝑏
𝐴𝑂 =−2𝑎+𝑏
𝐹𝐷 =−3𝑎+2𝑏 ? ?
𝒂= 𝒃= 3 − 𝒄= −3 0 𝒅= 𝒆= − 𝒇= −3 −4 ? ? ? ? ? ? ? ? ?

9. Starter
Expand and simplify the following expressions: ?
2 𝒂+𝒃 +𝒃 =2𝒂+3𝒃 𝒂−𝒃 +𝒃 = 1 2 𝒂+ 1 2 𝒃 𝒂 𝒂+𝒃 = 3 2 𝒂+ 1 2 𝒃 𝒂−2 𝒂−2𝒃 =−𝒂+4𝒃 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:
Find a relatively easy vector using skills from Lesson 1.
Find a harder vector that uses a fraction of your vector from part (a).
Edexcel June H Q27
Bro Tip: This ratio wasn’t in the original diagram. I like to add the ratio as a visual aid.
𝑆𝑄 =−𝒃+𝒂 ? 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).
3 : 2 ? ?
𝑁𝑅 = 2 5 𝑆𝑄 +𝒃 = 2 5 −𝒃+𝒂 +𝒃 = 2 5 𝒂+ 3 5 𝒃
𝑄𝑅 is also 𝑏 because it is exactly the same movement as 𝑃𝑆 . b
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.

11. Test Your Understanding
Edexcel June 2012
𝐴𝐵 = 𝐴𝑂 + 𝑂𝐵
=−𝒂+𝒃 ?
𝑂𝑃 =𝒂 𝐴𝐵
=𝒂 −𝒂+𝒃
= 1 4 𝒂+ 3 4 𝒃 ?
You MUST expand and simplify.

12. Further Test Your Understanding𝐵 𝐵 B
𝑂𝑋:𝑋𝐵=1:3
𝐴𝑌:𝑌𝐵=2:3 𝒃 𝑋 𝒃 𝑌 𝑂 𝒂 𝐴 𝑂 𝒂 𝐴
𝐴𝑋 =−𝒂 𝑂𝐵 =−𝒂 𝒂+𝒃 =−𝒂+ 1 4 𝒂+ 1 4 𝒃 =− 3 4 𝒂+ 1 4 𝒃
First Step?
𝑂𝑌 =𝒂 𝐴𝐵 =𝒂 −𝒂+𝒃 =𝒂− 2 5 𝒂+ 2 5 𝒃 = 3 5 𝒂+ 2 5 𝒃
First Step? ? ?

13. Exercise 2 ? ? ? ? ? ? ? ? ? (on provided sheet) 3 1
𝑋 is a point such that 𝐴𝑋:𝑋𝐵=1:4 3
𝐴𝐵 =−𝒂+𝒃
𝐴𝑋 =− 1 5 𝒂+ 1 5 𝒃
𝑂𝑋 = 4 5 𝒂+ 1 5 𝒃
𝐵𝑋 = 4 5 𝒂− 4 5 𝒃 ? ? ?
[June H 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: Show that 𝑂𝑃 = 𝒂+3𝒃 ? ? 2
𝑌 is a point such that 𝑌𝐵=2𝐴𝑌 ?
𝐴𝑌 =− 1 3 𝒂+ 1 3 𝒃
𝑂𝑌 = 2 3 𝒂+ 1 3 𝒃
𝑌𝑂 =− 2 3 𝒂− 1 3 𝒃 ? ? ?

14. Exercise 2 ? ? ? ? ? ? ? ? ? ? ? ? (on provided sheet)
[Nov H Q27] 𝑀 is the midpoint of 𝑂𝑃.
𝑂𝐴𝐶𝐵 is a parallelogram. 𝑅 is a point such that 𝐴𝑅:𝑅𝐵=2:3. 𝑆 is a point such that 𝐵𝑆:𝑆𝐶=1:3. 4 5 ?
𝑂𝑅 = 3 5 𝒂+ 2 5 𝒃
𝐵𝑆 = 1 4 𝒂
𝑂𝑆 = 1 4 𝒂+𝒃
𝑅𝑆 =− 7 20 𝒂+ 3 5 𝒃 ? ? ? 6
Express 𝑂𝑀 in terms of 𝒂 and 𝒃. 𝑂𝑀 = 1 2 𝒂+𝒃 𝑜𝑟 1 2 𝒂+ 1 2 𝒃
Express 𝑇𝑀 in terms of 𝒂 and 𝒃 giving your answer in its simplest form. 𝑇𝑀 =−𝒂 𝑂𝑀 =−𝒂 𝒂+𝒃 = 1 2 𝒃− 1 2 𝒂
𝑀 is the midpoint of 𝐶𝐷, 𝐵𝑃:𝑃𝑀=2:1 ?
𝐷𝐶 =−𝒛+𝒚
𝐷𝑀 =− 1 2 𝒛+ 1 2 𝒚
𝐴𝑀 = 𝐴𝐷 + 𝐷𝑀 = 1 2 𝒛+ 1 2 𝒚
𝐵𝑀 = 𝐵𝐴 + 𝐴𝑀 =−𝒙+ 1 2 𝒛+ 1 2 𝒚
𝐵𝑃 = 2 3 𝐵𝑀 =− 2 3 𝒙+ 1 3 𝒛+ 1 3 𝒚
𝐴𝑃 = 𝐴𝐵 + 𝐵𝑃 = 1 3 𝒙+ 1 3 𝒛+ 1 3 𝒚 ? ? ? ? ? ? ?

15. Exercise 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? (on provided sheet) 8 7
𝐴𝐵 =−𝒂+𝒃
𝐵𝐶 =𝒂
𝑀𝐵 =− 1 2 𝒂+ 1 2 𝒃
𝑀𝑋 =− 1 6 𝒂+ 1 2 𝒃
𝑋𝐴 = 2 3 𝒂−𝒃
𝐶𝑀 =− 1 2 𝒂− 1 2 𝒃
𝑋𝑂 =− 1 3 𝒂−𝒃 ?
𝑂𝐵 =𝒂+2𝒃
𝐵𝐶 =𝒂−𝒃
𝐴𝑀 = 1 2 𝒂+𝒃
𝑂𝑀 = 3 2 𝒂+𝒃 ? ? ? ? ? ? ? ? ? 9 ?
𝑂𝐶 =−𝒂+𝒃
𝑋𝐶 =− 3 4 𝒂+ 1 2 𝒃
𝐴𝑋 =− 5 4 𝒂+ 1 2 𝒃 ? ? ?

16. Recap ?

17. Types of vector ‘proof’ questions
“Show that … is parallel to …”
“Prove these two vectors are equal.”
“Prove that … is a straight line.”

18. Showing vectors are equal
Edexcel March 2013 ? ?
2 5 𝑂𝑌 = 𝑏+5𝑎−𝑏 =2𝑎+2𝑏
𝑂𝑋 =3𝑎 −3𝑎+6𝑏 =2𝑎+2𝑏
∴ 𝑂𝑋 = 2 5 𝑂𝑌
There is nothing mysterious about this kind of ‘prove’ question.
Just find any vectors involved, in this case, 𝑂𝑋 and 𝑂𝑌 . Hopefully they should be equal!
With proof questions you should restate the thing you are trying to prove, as a ‘conclusion’.

19. What do you notice? ? 𝒂+2𝒃 𝒂 2𝒂 2𝒂+4𝒃
! Vectors are parallel if one is a multiple of another. 𝒂
− 3 2 𝒂

20. Parallel or not parallel?
Vector 1
Vector 2
Parallel?
(in general)
𝑎+𝑏
𝑎+2𝑏
3𝑎+3𝑏
−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 show two vectors are parallel𝑨 𝑪 𝒂 𝑿 𝑴 𝑶 𝑩 𝒃
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 . ?
For any proof question always find the vectors involved first, in this case 𝑋𝑀 and 𝑂𝐶 .
𝑂𝐶 =𝑎+𝑏 𝑋𝑀 = 1 4 −𝑎+𝑏 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.
The magic words here are “is a multiple of”.

22. Test Your Understanding
Edexcel June 2011 Q26 𝐴
Find 𝐴𝐵 in terms of 𝒂 and 𝒃. −2𝒂+3𝒃
𝑃 is the point on 𝐴𝐵 such that 𝐴𝑃:𝑃𝐵=2:3. Show that 𝑂𝑃 is parallel to the vector 𝒂+𝒃. 𝑃 ? 2𝒂 𝐵 𝑂 3𝒃 ?
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 points form 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. ? C B A C B A

24. Straight Line Example ? ? a −3𝒃+𝒂 b 𝐴𝑁 =2𝒃, 𝑁𝑃 =𝒃
𝑁𝑀 =𝑏 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
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.

25. Test Your Understanding
November H Q24
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. ? ?

26. Exercise 3 1 2
𝑇 is the point on 𝐴𝐵 such that 𝐴𝑇:𝑇𝐵=5:1. Show that 𝑂𝑇 is parallel to the vector 𝒂+2𝒃.
𝑂𝑇 =2𝒃 −2𝒃+5𝒂 =2𝒃− 1 3 𝒃+ 5 6 𝒂 = 5 6 𝒂+ 5 3 𝒃 = 5 6 𝒂+2𝒃
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. ? ?

27. Exercise 3 3 4
𝐴𝐶𝐸𝐹 is a parallelogram. 𝐵 is the midpoint of 𝐴𝐶. 𝑀 is the midpoint of 𝐵𝐸. Show that 𝐴𝑀𝐷 is a straight line.
𝐷𝑀 =−𝒃 𝒃+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.
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 𝒂− 1 2 𝒃 = 3 5 𝒂+ 3 5 𝒃 = 3 5 𝒂+𝒃 𝐶𝐸 =𝒂+𝒃 ? ? ? ?

28. Exercise 3 5 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 −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.
𝑂𝐴𝐵𝐶 is a 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 −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. ? ? ?