1. CIE Centre A-level Pure Maths
P1 Chapter 13
CIE Centre A-level Pure Maths
© Adam Gibson

2. scalars
matrices
vectors
operators
geometrical
objects
tensors

3. VECTORS
A vector is a mathematical object which has two properties:
SIZE
&
direction
Examples of vector quantities:
velocity, force, distance, temperature, torque

4. Vectors in 2D – the basics
We can write a vector in the x-y plane as a certain number
of x-units and a certain number of y-units.
When you write
vectors, you should
underline them 4 3
This vector is specified by 3 units in the x-direction and 4 units in the y-direction.
We can write it as
or as
; 3 and 4 are the components
In the x- and y-directions.

5. Vectors in 2D – the basics - magnitude
The magnitude of a vector is another name for its size.
What is the size
of this vector? 4 3
It can be calculated using Pythagoras’ theorem.
The size is the same as the length,
which is

6. Vectors in 2D – the basics - location?
Question: Where is the vector
Answer: everywhere!
A vector can represent a simple translation

7. Unit vectors
A unit vector can be
calculated in any direction
by dividing by the magnitude.
1 unit
1 unit
Every vector in the x-y
plane can be expressed as
a linear combination of i and j.
Write the vector from (2,7) to (8,-5) in terms of i and j.
Calculate the unit vector in the direction of
the vector

8. Vector arithmetic
The rules for adding and subtracting vectors are easy..
The Parallelogram Rule…
Now look at Fig 13.5
p. 191
means that

9. Vector arithmetic – continued.
Hence you should see that:
“the commutative rule for the addition of vectors is a
consequence of the commutative rule for the addition of
real numbers”. If
, what is the magnitude of p + q ?
Subtraction is no different:
Calculate the magnitude of
and illustrate your
calculation on a graph.

10. Vector arithmetic – continued.
Q: We can add and subtract vectors, but can we multiply
or divide them?
The answer is yes and no.
Multiplication by a scalar.
Illustrate, on a graph, the meaning of
How is the picture different if s < 0 or s > 0?
Now try Q 3, 5 and 10
on page 194.

11. Vectors – 3D
Vectors in 3 dimensions are very useful for solving
real world (or virtual world!) problems.

12. 3D vs 2D compared.
Q: Consider everything we’ve learnt about lines, equations
and calculus. Is it different in a 3 dimensional coordinate
system?
Non-parallel lines don’t always cross. Such lines are
called skew.
Gradient – it’s not defined purely by a number. The
direction of a line is only defined by a vector.
Planes in 3D are the equivalent of lines in 2D. Non-parallel
planes always do cross. We will study how to represent
planes in vector form later.

13. The dot product gives a quantitative measure
The “scalar product” or “dot product”
The scalar product is defined, for any 2 vectors p and q,
as:
Why do we call it a “scalar” product?
What is |p|?
What is the angle θ?
If x=24 and p=3i+2j, what is x.p?
What is the real meaning of the dot product?
The dot product gives a quantitative measure
of the extent to which 2 vectors are collinear.

14. The “scalar product” or “dot product” - continued
The dot product a.b is..
zero b a b
The dot product a.b is..
|a||b| a
If |a|=|b|=1, then
the dot product a.b is.. b
0.5 a

15. The “scalar product” or “dot product” - continued
What does it mean if p.q = 0?
The vectors are
perpendicular.
What is p.p?
It’s the square of the magnitude
Of the vector, |p|2
In component form
Assume the distributive rule:
See p. 205 –
you don’t need to prove this
Then remembering i.i = j.j = k.k = 1 and
i.j = j.k = i.k = 0,

16. The “scalar product” or “dot product” - continued
We thereby get the very useful result:
Which means we don’t need
to use trigonometric functions
to calculate dot products.
In particular we can see that a.a = |a|2 implies:
(note that this can be extended to
any number of dimensions!)

17. TASKS
Find the magnitude of each of the following vectors.
Which vectors are perpendicular?
What is the angle between p and q (in radians, 1d.p.)?
**Can you find a vector skew to all these four?
A: |p| = √14, |q|=√369/2,|r|= √19,s=2 √5
p and r are perpendicular, p and s are perpendicular,
Angle between p and q is: cos = 1.0 radian

18. Relative Velocity – Key Principles
Know how to resolve vectors
Know and understand the key equation for relative
velocity:
“The velocity of b relative to a is the velocity of b minus the velocity of a”

19. Problems with Relative VelocityN W E S
New York
5600km
London
A plane flies directly West from London to New York, a distance of 5600km. Its maximum speed through still air is 950 km/hr. The wind speed is 120 km/hr at a bearing of 20 degrees. How quickly can it reach New York, and at what bearing does the plane fly relative to the air?

20. Problems with Relative VelocityN W E S
New York
5600km
London

21. Problems with Relative Velocity
The velocity of the plane relative to the air is the true velocity of the plane minus the velocity of the air
land
water speed
It’s the same for a boat. The boat’s direction in the water must be towards the left (opposing the water) if we want to travel directly to the other side.

22. Problems with Relative Velocity
We need to resolve the wind
(air) speed into two components. 20
The aircraft must travel due West. So what is ?

23. Problems with Relative Velocity
We know that the magnitude of is 950.
So we can draw like this:

24. Problems with Relative Velocity
To make the problem simpler, we define unit vectors i and j in the directions West and North respectively.

25. Problems with Relative Velocity
At what angle is the plane flying relative to the air? N
Bearing =

26. Problems with Relative Velocity
A traveller on a train travelling with velocity
sees a car out of the window, which appears to be travelling at velocity
What is the true speed of the car?
True velocity of car is given by: