IGCSE – Vectors
Dr C Liu (CKL@whitgift.co.uk)
IGCSE Specifications:
A02 5.1A to G (vectors)
Last modified: 20 Apr 21
IGCSE Specifications
Coordinates vs Vectors
𝑦
6 ?
Coordinates represent a position.
We write to indicate the position
5
and position.
4
( 2,3?) Vectors represent a ?
3 movement. We write to
2 ()
4
1?
indicate the change in and
change in .
Note we put the numbers
1 vertically. Do NOT write ; there
is no line and vectors are very
1 2 3 4 5 6 7
𝑥 different to fractions!
-1 You may have seen vectors
briefly before if you’ve done
transformations; we can use
-2 them to describe the
movement of a shape in a
translation.
Vectors to represent movements
𝒃
𝒂
1 5
() ()
? ?
𝒅
𝒇 𝒂= 𝒃= ? ?
𝒄
𝒆 𝒈 3 1 ?
?
?
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
𝒃 variables for vectors?
We use capital letters for vertices
𝑍 ? for vectors.
and lower case letters
There’s 3 ways in which can
𝒂 represent the vector from point to :
1. (in bold)
2.
?
(with an ‘underbar’)
3. ?
𝑋 ?
Adding Vectors
Follow the ?
Path! ?
𝑌 ?
𝒃 What do you notice about the numbers in
when compared to and ?
𝑍 We’ve simply added the values and values to
describe the combined movement.
i.e.
𝒂 ?
NB: we can use any route to get from the start to
finish, and the resultant vector will always be the
𝑋 same.
• Route 1: We go from to via .
• Route 2: Use the direct line from to :
Scaling Vectors
We can ‘scale’ a vector by
multiplying it by a normal number,
aptly known as a scalar.
If , then
?
What is the same about and and
what is different?
𝒂
2 Same:
Same direction / Parallel
?
Different:
The length of the vector, known as
the magnitude, is?longer.
Note that vector letters are
bold but scalars are not.
More on Adding/Subtracting Vectors
𝑋 2𝒄
If , and , then find the following
in terms of , and :
𝐴 𝐵
𝒂 𝒃 𝒃 𝑌 ?
?
𝒂
?
𝑂 𝒃 ?
?
Side Note: Since would Note: Since , subtracting a
end up at the same finish vector goes in the opposite
point, we can see (i.e. direction.
vector addition, like
normal addition, is
‘commutative’)
Exercise 1 (on provided sheet)
1 State the value of each vector. 2 4
𝒂 𝒃
a. ? a.
?
𝒄 b.
? b.
?
?
c. c.
d. d. ?
? ?
𝒅 𝒆
3 5
𝒇
a. ?
b.
? a. ?
c.
d. ? b. ?
? c.
?
() ( ) ( )
3?
𝒂= 𝒃=
1?
3?
−?2
𝒄=
−? 3
?0
d.
?
Starter
Expand and simplify the following expressions:
2(𝒂+𝒃)+𝒃=2𝒂+3𝒃
1 ?
2 ?
3 ?
4 ?
5 ?
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
Tip: This ratio
wasn’t in the
original
a ⃗ ? 𝒂
𝑆𝑄=− 𝒃+
diagram. I like
to add the ratio 3 For (b), there’s two possible paths to
as a visual aid. :
2 get from to : via or via . But which
is best?
In (a) we found to rather than to ,
so it makes sense to go in this
direction so that?we can use our
result in (a).
2
is also because it is
b
⃗ ⃗
exactly the same
𝑁𝑅= 𝑆𝑄+𝒃
movement as .
?
5
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.
Test Your Understanding
Edexcel June 2012
You MUST
? expand and
simplify.
Further Test Your Understanding
A
𝑂𝑋 : 𝑋𝐵=1:3 𝐵 B 𝐴𝑌 :𝑌𝐵=2 :3 𝐵
𝒃
𝑋 𝒃
𝑌
𝑂 𝒂 𝐴 𝑂 𝒂 𝐴
First Step?
⃗ 2 First Step?
⃗
𝑂𝑌=𝒂+ 𝐴𝐵
?
5 ?
Exercise 2 (on provided sheet)
1 is a point such that 3
a. ?
b.
c.
?
d. ? [June 2009 2H Q23]
? a) Find in terms of and .
2 is a point such
?
b) is on such that .
that Show that
a.
b.
?
c. ?
? ?
Exercise 2 (on provided sheet)
4 [Nov 2010 1H Q27] is the midpoint 5
is a parallelogram. is a
point such that . is a point
of .
such that .
a. ?
b.
c. ?
d. ?
?
6
a) Express in terms of and .
is the midpoint of ,
?
b) Express in terms of and giving
your answer in its simplest form. a. ?
b.
c.
?
d. ?
e. ?
? f.
?
?
Exercise 2 (on provided sheet)
7 8
?
? a. ?
b.
? c. ?
? d. ?
? ?
? 9
?
?
?
?
Your Go …
!
Recap of Vectors (… leading to Proofs)
?
?
?
?
Types of vector ‘proof’ questions
“Show that … is
parallel to …”
“Prove these two
vectors are equal.”
“Prove that … is
a straight line.”
Showing vectors are equal
Edexcel March 2013
?
There is nothing
mysterious about this
kind of ‘prove’ question. Q.E.D.
Just find any vectors ?
involved, in this case, With proof questions
and . Hopefully they you should restate the
should be equal! thing you are trying to
? prove, as a ‘conclusion’.
What do you notice?
𝒂 2𝒂 𝒂 +2 𝒃
2 𝒂+4 𝒃
! Vectors are parallel
𝒂 3 ?
if one is a multiple of
− 𝒂
2 another.
Parallel or not parallel?
Vector 1 Vector 2 Parallel?
(in general)
No
Yes
No
Yes
No
Yes
No
Yes
For ones which are parallel, show it diagrammatically.
How to show two vectors are parallel (
𝑨 𝑪
𝒂 𝑿 𝑴
𝑶 𝑩
𝒃
is a point on such that . is the midpoint of .
Show that is parallel to .
For any proof question always find the vectors
? involved first, in this case and .
is a multiple of parallel
i.e.
? The key is to factor out a scalar such
that we see the same vector.
? The magic words here are
“is a multiple of”.
Test Your Understanding
Edexcel June 2011 Q26
𝐴 a) Find in terms of and .
𝑃
2𝒂 ?
𝐵 b) is the point on such that .
Show that is parallel to the vector .
𝑂 3𝒃
?
?
?
NB: the mark scheme didn’t specifically
require “is a multiple of” here (but write it
anyway!), but DID explicitly factorise out the
Proving three points form a straight line (collinear)
Points A, B and C form a straight line if:
and are parallel (and B is? a common point).
Alternatively, we could show and are parallel. This tends to be easier.
C
B
A
C
B
A
Straight Line Example
a
−3 𝒃+ 𝒂?
b
?
is a multiple of is parallel to .
?
is a common point.
?
is a straight line.
is the midpoint of . is the midpoint of . ?
a) Find in terms of and .
b) Show that is a straight line.
Test Your Understanding
November 2013 1H Q24
and
is the point such that
The point divides in the ratio .
(a) Write an expression for in terms of and .
?
(b) Prove that is a straight line.
is a multiple of and is a common point.
is a straight line. ?
?
Exercise 3
1 2
is the point on such that . Show is the midpoint of . is the midpoint of .
that is parallel to the vector . Prove that is parallel to .
is a multiple of parallel.
?
?
Exercise 3
3 4
is a parallelogram. is the midpoint of . is , and
the midpoint of . Show that is a straight i) Express in terms of and .
line.
?
ii) Prove that is parallel to .
which
is a multiple of and is a common point, ?
is a multiple of
so is a straight line. iii) is the point on such that such that . Prove that ,
and lie on the same straight line.
?
Exercise 3
5 6
and and and . is the midpoint of . Prove
is a parallelogram. is the point on such
that . that is a straight line.
i) Find the vector . Give your answer in
terms of and . is a multiple of and is a common point,
therefore is a straight line.
?
ii) Given that the midpoint of is , prove
that is a straight line.
?
is a multiple of and is a common point,
therefore is a straight line.
?
Appendix: Solutions …
Exercise 2 (on provided sheet)
1 is a point such that 3
a. ?
b.
c.
?
d. ? [June 2009 2H Q23]
? a) Find in terms of and .
2 is a point such
?
b) is on such that .
that Show that
a.
b.
?
c. ?
? ?
Exercise 2 (on provided sheet)
4 [Nov 2010 1H Q27] is the midpoint 5
is a parallelogram. is a
point such that . is a point
of .
such that .
a. ?
b.
c. ?
d. ?
?
6
a) Express in terms of and .
is the midpoint of ,
?
b) Express in terms of and giving
your answer in its simplest form. a. ?
b.
c.
?
d. ?
e. ?
? f.
?
?
Exercise 2 (on provided sheet)
7 8
?
? a. ?
b.
? c. ?
? d. ?
? ?
? 9
?
?
?
?
Exercise 3 (on provided sheet)
3 4
is a parallelogram. is the midpoint of . is , and
the midpoint of . Show that is a straight i) Express in terms of and .
line.
?
ii) Prove that is parallel to .
which
is a multiple of and is a common point, ?
is a multiple of
so is a straight line. iii) is the point on such that such that . Prove that ,
and lie on the same straight line.
