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Mr Barton’s Maths Notes
Shape and Space
9. Vectors
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9. Vectors
1. What are Vectors?
• Vectors are just a posh (and quite
convenient) way of describing how to get
from one point to another
• Starting from the tail of the vector,
the number on the top tells you how far
right/left to go, and the number on the
bottom tells your how far up/down
b
a
c
d
If this number is positive,
you move right, if it is
negative, you move left
3
 
 4
a
b
c
d
If this number is positive,
you move up, if it is
negative, you move down
1 
 
 3
5 
 
 2 
 3 
 
 2 
0
 
3
1 to the right, and 3 up
5 to the right, and 2 down
3 to the left, and 2 down
0 to the right, and 3 up
2. The Magnitude of Vectors
• By forming right-angled triangles and using Pythagoras’ Theorem, it is possible to work out
the magnitude (size) of any vector
5
a
2
b
3
4
a
b
5 
 
 2 
3
 
 4
a 2  52  22
a  52  22
a  5.4 (1 dp)
b2  32  42
b  32  42
b5
Note: Because you are squaring the numbers,
you do not need to worrying about negatives!
3. Adding Vectors
• When you add two or more vectors together, you simply add the tops and add the bottoms
• The new vector you end up with is called the resultant vector
b
a
a+b
d
 3
b  4
 
 
3
 
1 
 3  4   7 
a+b
     
 3  1   4 
a
c+d
c
c
 5 
 
2 
c+d
Watch Out! Remember to be careful with your
negatives!
d
 2 
 
4 
 5   2   7 
     
2  4  6 
4. Subtracting Vectors
• The negative of a vector goes in the exact opposite direction, which changes the signs of the
numbers on the top and the bottom (see below)
• One way to think about subtracting vectors is to simply add the negative of the vector!
a
-a
4 
 
 2 
-a
 4 
 
2 
a
p
-q
q
 2
 
 4
q
3
 
 2
p
p
p-q
p–q
=
p +
(-q)
 2   3   1
     
 4   2   2 
5. Multiplying Vectors
• The only thing you need to remember when multiplying vectors is that you multiply both the
top and the bottom of the vector!
p
p
 2
 
 2
q
3
 
 2
r
1 
 
 2
2p
q
3q
r
- 4r
2p
 2  4
2    
 2  4
3q
3 9
3    
 2  6
-4r
1   4 
4     
 2   8 
6. Linear Combinations of Vectors
• Using the skills we learnt when multiplying vectors, it is possible to calculate some pretty
complicated looking combinations of vectors
Example: If
 3
a 
 5
 4 
b 
2 
(a) 4a + 3b + c
3
 4 
 1 
4    3    
5
2 
 2 
12 
 12 
 1 
   
   
 20 
6 
 2 
 1 
 
 24 
 1 
c 
 2 
Calculate the following:
(a) 2a - 5b - 2c
3
 4 
 1 
2   5   2 
5
2 
 2 
6 
 20 
 2 
   
   
10 
10 
 4 
 28 
 
4 
Watch Out! Remember to be so, so careful with your negatives!
7. Vectors in Geometry
• A popular question asked by the lovely examiners is to give you a shape and ask you to
describe a route between two points using vectors.
• There is one absolutely crucial rule here… you can only travel along a route of known vectors!
Just because a line looks like it should be a certain vector, doesn’t mean it is!
Example: Below is a regular hexagon. Describe the routes given in terms of vectors a and b

(i) FC
a
B
A
The best way to go here is straight across
the middle, because we know each horizontal
line is just a

b
FC
O
F
= 2a
C

(ii) DA
E
D
Again, the middle is looking good here, but
remember we are going the opposite way to
our given vector, so we need the negative!

DA
= – 2b

(iii) EB
a
B
A
It would be nice to just nip across the
middle, but the problem is we do not know
what those vectors are! So… we’ll just have
to go the long way around, travelling along
routes we do know!
b
O
F
C





EB = EF + FO + OA + AB
= -b +
a
= 2a – 2b
E
D
+ -b + a
Good luck with
your revision!