1. Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com
GCSE: Vectors
Dr J Frost
Last modified: 31st August 2015

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

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

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

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

6. Combining Portions of Vectors𝐶 𝑃 𝑏 𝐴 𝑎 𝑌 𝐵 𝑋
Let 𝐴𝐵 =𝑎 and 𝐵𝐶 =𝑏
Let 𝑋 be the midpoint of 𝐴𝐵, 𝑌 be a point on 𝐵𝐶 such that 𝐵𝑌:𝑌𝐶=1:3 and 𝑃 be a point on 𝐴𝐶 such that 𝐴𝑃:𝑃𝐶=2:1
Find: ?
𝐴𝑋 = 𝟏 𝟐 𝒂 𝐴𝑌 =𝒂+ 𝟏 𝟒 𝒃 𝐶𝑌 =− 𝟑 𝟒 𝒃 ?
𝐴𝐶 =𝒂+𝒃 𝐴𝑃 = 𝟐 𝟑 𝑨𝑪 = 𝟐 𝟑 𝒂+ 𝟐 𝟑 𝒃 ? ? ?

7. Exercise 1 (on your sheet)3
𝐵𝐴 =−𝑎
𝐴𝐶 =𝑎+𝑏
𝐷𝐵 =2𝑎−𝑏
𝐴𝐷 =−𝑎+𝑏 ?
𝑍𝑋 =𝑎+𝑏
𝑌𝑊 =𝑎−2𝑏
𝑋𝑌 −𝑎+𝑏
𝑋𝑍 =−𝑎−𝑏 ? ? ? ? ? ? ? 2 4
𝑀𝐾 =−𝑎−𝑏
𝑁𝐿 =3𝑎−𝑏
𝑁𝐾 =2𝑎−𝑏
𝐾𝑁 =−2𝑎+𝑏 ? ?
𝐴𝐵 =−𝑎+𝑏
𝐹𝑂 =−𝑎+𝑏
𝐴𝑂 =−2𝑎+𝑏
𝐹𝐷 =−3𝑎+2𝑏 ? ? ? ? ? ?

8. Quick fire ratio A P B z ? ? ? ? ? ? The vector 𝒛= 𝐴𝐵
Find the following vectors given the specified ratios: 𝐴𝑃 𝐵𝑃 ?
1 2 𝒛
− 1 2 𝒛 ?
𝐴𝑃:𝑃𝐵 = 1:1 ?
1 3 𝒛 ?
− 2 3 𝒛
𝐴𝑃:𝑃𝐵 =1:2 ?
3 7 𝒛 ?
− 4 3 𝒛
𝐴𝑃:𝑃𝐵 =3:4

9. More Complex Paths ? ? 𝐴𝐵 = 𝐴𝑂 + 𝑂𝐵 =−𝒂+𝒃 𝑂𝑃 =𝒂+ 3 4 𝐴𝐵 =𝒂+ 3 4 −𝒂+𝒃
Bro Tip: GCSE vectors questions will almost always be in two parts: Part (b) will use your answer from part (a).
𝐴𝐵 = 𝐴𝑂 + 𝑂𝐵
=−𝒂+𝒃 ?
𝑂𝑃 =𝒂 𝐴𝐵
=𝒂 −𝒂+𝒃
= 1 4 𝒂+ 3 4 𝒃 ?
You MUST expand and simplify.

10. Test Your Understanding1 𝐵 𝐵 2
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? ? ?

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

12. Given that M is the midpoint of BC, determine AM.
3a + 2b
3a + b
2a + 3b
a + b

13. Exercise 2 (on your sheet)3 1
𝐴𝐵 =−𝑎+𝑏
𝐴𝑋 =− 1 5 𝑎+ 1 5 𝑏
𝑂𝑋 = 4 5 𝑎+ 1 5 𝑏
𝐵𝑋 = 4 5 𝑎− 4 5 𝑏 ?
𝑂𝑅 =𝑎 −𝑎+𝑏 = 3 5 𝑎+ 2 5 𝑏
𝐵𝑆 = 1 4 𝑎
𝑂𝑆 = 1 4 𝑎+𝑏
𝑅𝑆 = 3 5 −𝑎+𝑏 𝑎 =− 7 20 𝑎+ 3 5 𝑏 ? ? ? ? ? ? ? 2 4
𝐴𝑌 =− 1 3 𝑎+ 1 3 𝑏
𝑂𝑌 = 2 3 𝑎+ 1 3 𝑏
𝑌𝑂 =− 2 3 𝑎− 1 3 𝑏 ? ?
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 ? ? ? ? ? ? ?

14. Recap ?

15. Dealing with more complicated ‘routes’
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
The key is to choose a suitable ‘route’ and work out each part separately before adding. Sometimes there are multiple possible routes.
But try to use vectors you have found in previous parts of the question.

16. Test Your Understanding𝑨 𝑪 𝒂 𝑴 𝑿 𝑶 𝑩 𝒃
Given that 𝑋 is a point such that 𝐴𝑋:𝑋𝐵=3:1 and 𝑀 is the midpoint of 𝐵𝐶, find:
𝐴𝐵 =−𝒂+𝒃
𝑋𝑀 = 𝟏 𝟒 −𝒂+𝒃 + 𝟏 𝟐 𝒂 = 𝟏 𝟒 𝒂+ 𝟏 𝟒 𝒃 ? ?

17. Exercise 3 (on sheet) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 𝑂𝐵 =𝑎+2𝑏 1
𝐵𝐶 =𝑎−𝑏
𝐴𝑀 = 1 2 𝑎+𝑏
𝑂𝑀 = 3 2 𝑎+𝑏 4 ? ?
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 ? ? ?
𝑂𝐶 =−𝑎+𝑏
𝑋𝐶 =− 3 4 𝑎+ 1 2 𝑏
𝐴𝑋 =− 5 4 𝑎+ 1 2 𝑏 ? ? 3 ? ? ? ?
𝑋𝑌 = 1 2 𝑎+ 1 3 𝑏
𝑌𝑍 = 2 3 𝑎+ 1 4 𝑐
𝑋𝑍 = 1 2 𝑎+𝑏+ 1 4 𝑐
𝑂𝑍 =𝑎+𝑏+ 1 4 𝑐
𝐶𝑋 =−𝑐−𝑏− 1 2 𝑎 ? ? ? ? ? 2 ? ?

18. What do you notice? ? 𝒂+2𝒃 𝒂 2𝒂 2𝒂+4𝒃
! Vectors are parallel if they have the same direction (but possibly different magnitudes). 𝒂
− 3 2 𝒂

19. 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.

20. How to show two vectors are parallel𝑨 𝑪 𝒂 𝑴 𝑿 𝑶 𝑩 𝒃
We earlier found that 𝑋𝑀 = 𝟏 𝟒 𝒂+ 𝟏 𝟒 𝒃
How do we show this is parallel to OC ? ?
! To show parallel, factor out scalars so same vector in brackets:
𝑋𝑀 = 1 4 𝑎+𝑏 Then write “ 𝑋𝑀 is a multiple of 𝑂𝐶 ”

21. Test Your Understanding? ?

22. 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. ? C B A C B A

23. Test Your Understanding
𝑁𝑀 =𝑏 𝑎−3𝑏 𝑀𝐶 = 1 2 𝑎−3𝑏 +𝑎 Simplify to 𝑎−𝑏 and 𝑎−𝑏
“NM is a multiple of MC”
(+ they have a common point M)
1 mark? b
1 mark?
1 mark?
1 mark?
−3𝒃+𝒂
1 mark? a

24. Exercise 4 1
𝐶𝐸 =𝒂+𝒃
𝐹𝐸 =𝑎−𝑏+𝑎+𝑏=2𝑎, 𝐶𝐷=𝑎. 𝐹𝐸 is a multiple of 𝐶𝐷.
𝐹𝑀 =𝑎−𝑏+𝑎+ 1 2 𝑏=2𝑎− 1 2 𝑏
𝐶𝑋 = 𝐶𝐹 + 𝐹𝑋 =𝑏−𝑎 𝑎− 1 2 𝑏 = 3 5 𝑏+ 3 5 𝑎= 𝑎+𝑏 𝑋𝐸 = 𝑋𝑀 + 𝑀𝐸 = 𝑎− 1 2 𝑏 𝑏= 2 5 𝑎 𝑏= 2 5 (𝑎+𝑏) 𝐶𝑋 is a multiple of 𝑋𝐸 , so parallel. They share a common point 𝑋.
i) 𝑂𝑃 = 𝑂𝐴 + 𝐴𝑃
=6𝑎 −6𝑎+6𝑐 =2𝑎+4𝑐 =2(𝑎+2𝑐)
ii) 𝑃𝑀 = 𝑃𝐶 𝐶𝐵 = 1 3 −6𝑎+6𝑐 +3𝑎=𝑎+2𝑐 𝑃𝑀 is a multiple of 𝑂𝑃 , so they are parallel. They also share a common point 𝑃. ? 3 ?
𝐴𝑀 =𝑎 −4𝑎+𝑏 =−𝑎+ 1 2 𝑏 𝑀𝐶 = 1 2 −4𝑎+𝑏 𝑏 =−2𝑎+𝑏 =2 −𝑎+ 1 2 𝑏
𝐴𝑀 is a multiple of 𝑀𝐶 (and 𝑀 is a common point). ? ? ? 2 ? ?

25. Answers on next slides. GCSE Questions
File Ref: GCSERevision-Vectors.docx

26. << Return to Index
Vectors
<< Return to Index ? ?

27. << Return to Index
Vectors
<< Return to Index ? ?

28. << Return to Index
Vectors
<< Return to Index ? ?

29. << Return to Index
Vectors
<< Return to Index ? ?

30. << Return to Index
Vectors
<< Return to Index ? ?

31. << Return to Index
Vectors
<< Return to Index ? ?

32. << Return to Index
Vectors
<< Return to Index ? ?

33. << Return to Index
Vectors
<< Return to Index ? ?