Vectors

Forces

Vectors & Scalars

9U
Acceleration
Vectors Engage

Two cars are driving down a single-track lane at 30 miles

per hour
What might happen?

Which of these possibilities did you identify?

Both travelling in the Both travelling away from Both travelling towards
same direction each other each other
 Vectors Challenge & Develop

This Cars question nicely demonstrates that sometimes our language

lacks detail.
If you were standing in a field beside that lane looking at the cars,
you would be able to immediately spot whether the cars were going
in the same direction, or not.
Working with only their speed, you can not definitely state what will
happen to the cars.
This is because the information you were given only had magnitude
(size), and nothing relating to the direction.
Speed is an example of a scalar quantity – something which can be
measured with numbers & units
 Vectors Challenge & Develop

Suppose the question had looked like this instead;

The resultant forces acting on two identical boats are shown below;

10 N

1
0
N
Describe what will happen to each boat

What will happen to each boat?

 Vectors Challenge & Develop

In the example of the boats, the force-arrows let you predict the 10 N
fate of each boat because they told you both magnitude (size)
and direction.

In words; it could have been written as

1
“a force of 10 N downwards” or “10 N to the left”.
0
N
This is an example of a vector quantity.

In maths you may have met vectors which look like this B
diagram 
It’s the same idea – measuring the length of the line gives
you the magnitude (size), and the angle shows you the
direction A
Vectors Challenge & Develop

Scalar quantities have magnitude (size) only.

For example;
● Temperature

Vector quantities have both magnitude and an associated direction.

Examples include;
● Force

A vector quantity may be represented by an arrow. The length of the

arrow represents the magnitude, and the direction of the arrow the
direction of the vector quantity.

What other examples of Scalars & Vectors can you

suggest?
Vectors Challenge & Develop

Remember!
When we say that a vector has a direction,
there are two possibilities.

Either, the quantity itself is to do with movement

e.g. the velocity of a car
or
the quantity is capable of creating a movement in something else
e.g. a force, or a magnetic field.
 Vectors Challenge & Develop

For each quantity, decide if it is a Scalar or a Vector

2200
16
N down
102 cmC
miles
o
Scalars Vectors
Northwest
10 7m/s
s2
Length – e.g. Force – e.g.

Temperature Acceleration
- e.g. – e.g.
Time – e.g. Displacement
- e.g.
Vectors Use your existing knowledge to sort these Challenge & Develop
quantities into Scalars or Vectors
Scalar Force Vector
Temperature
Speed
Displacement
Distance
Power
Velocity
Acceleration
Pressure
Wavelength
Magnetic Field
Mass
Current
Momentum
 Vectors Explain

Speed is a measure of how far an object moves in a given time.

This car is travelling at 60 mph.

This means the car travels 60 miles every hour.

This jet is travelling at 350 m/s.

This means the jet travels 350 metres every second.

Is Speed a Scalar or a Vector?

Vectors Explain

The speed of an object does not depend on the

direction in which it is travelling.

The velocity of an object is the speed and

direction in which it is moving.

The car is travelling north at 10 m/s.

What happens to the Velocity as the car

goes round the corner?

As the car goes round the corner, the speed of the car
remains constant but the velocity changes because the
direction has changed!
 Vectors Explain

Velocity is a vector because it has both magnitude and direction.

It changes whenever there is;

● a change in its magnitude (i.e. a change in speed)
or
● a change in its direction
or
● both

When identifying the velocity of different objects, we choose one direction to be positive –
the opposite direction is therefore negative!
For example; both these cars are travelling at a speed of 3 m/s, but their velocities are
opposite to each other!

3 m/s -3 m/s
 Vectors Explain

Revisiting the Cars question - Two cars are driving down a single-
track lane. The red car is travelling at a velocity of 3 m/s; the blue
car at -3 m/s.

What might happen?

Which of these possibilities did you identify?

Both travelling away from Both travelling towards

each other each other
 Vectors Explain

Imagine Bob is standing a point A; B

A 100m

He wants to get to point B, which is 100m away.

If he simply closes his eyes and walks 100m

from A – will he arrive at Point B?
 Vectors Explain

Just walking a distance of 100m will not get Bob to Not B

Point B. B

To get to point B successfully, Bob needs to walk 100m

100m in a particular direction – this is an example
of the vector quantity called displacement. Still Not B
100m
100m A

100m
Not B either

Definitely Not B
 Vectors Explain

Harry and Sally are exploring the desert.

They need to reach an oasis, but choose to take different routes.

 Harry travels due north, then due east.

 Sally simply travels in a straight line to the oasis.

Did they both travel the same distance?

When Harry met Sally at the oasis, they had travelled different
distances. However, because they both reached the same
destination from the same starting point, their overall
displacements were the same.
 Vectors Explain

Distance is how far an object moves, but it does not involve

direction.
Distance is a scalar quantity.

Displacement includes both the distance an object moves,

measured in a straight line from the start point to the finish point
and the direction of that straight line.
Displacement is a vector quantity.

Consider this scenario;

A woman goes into her local library and walks straight ahead for 20m before turning
right to enter the stacks of shelves. She turns right again after 5m and stops after
another 8m.

How far has she walked from the entrance?

Vectors Explain
A woman goes into her local library and walks straight ahead
for 20m before turning right to enter the stacks of shelves. She
turns right again after 5m and stops after another 8m.

If you look at the question in scalar quantities, she has

travelled a distance of 33 m from the door.
20 m + 5 m + 8 m = 33 m

However, if you look at the arrows, she actually is only 13 m

from the door. Her displacement is 13 m at an angle of 22.6o
from the door.
(Alternatively, we could say that her displacement from the start point to the
end point is 13m)
 Vectors Explain

Imagine a plane flying a ‘loop-the-loop’

What happens to the;

● speed…
● velocity…
● distance travelled…
● displacement…
over the course of the loop?

START &
END POINT
Vectors Consolidate & Apply

Complete the questions from the sheet

in your books
Vectors Consolidate & Apply

Question 1 Question 1
State whether the following are examples of vector or scalar quantities: a) Scalar
a) A distance of 2 m b) Vector
b) A force of 35 N acting to the right
c) Vector
c) A velocity of 3 m/s on a bearing of 115o
d) Scalar
d) 55 s
e) Scalar
e) 4 m/s
f) Vector
f) 165m North
 
Question 2
Question 2
How long should a line be to represent 19 N if you use a scale of:
g) 9.5 cm
g) 1 cm: 2 N
h) 6.33 cm
h) 1 cm: 3 N i) 3.8 cm
i) 1 cm: 5 N
 Vectors Consolidate & Apply

Question 3 Question 3
a) Find the total distance travelled from A to B a) 13 km
to C to D to E in this journey
b) 1.5 km south

Question 4
The resultant of a number of forces is the
single force that will have the same effect as
the original forces acting together.
b) What is the displacement in the journey
given above, if the start of the journey was
going East?

Question 4
Give the definition for ‘resultant force’.
 Vectors Consolidate & Apply

Question 5 Question 5
Taking right as the positive direction, find a) Resultant = (+2) + (+7) = +9
the resultant of the following forces: (You so 9 N to the right
may find drawing a diagram in each case
helpful)
b) Resultant = (+4) + (-12) = +8
a) 2 N right; 7 N right
so 8 N to the left
b) 4 N right; 12 N left
c) 12 N left; 16 N left
c) Resultant = (-12) + (-16) = -28
d) 3 N right; 6 N left; 15 N right; 8 N left
so 28 N to the left
e) 16 N left; 24 N right; 13 N left
d) Resultant = (+3) + (-6) + (+15) + (-8) = +4
so 4 N to the right

e) Resultant = (-16) + (+24) + (-13) = -5

so 5 N to the left
 Vectors Extend

Three Cars are using a roundabout.

What happens to the;

● speed…
● velocity…
● distance travelled…
● displacement…
of each car?
Vectors
Forces

Vector Diagrams

9U
Acceleration
Vectors Engage

Predict the outcome of each Tug o’ War…

What additional information do you need?

 Vectors Challenge & Develop

As seen previously, a force is a vector quantity; meaning a force arrow can provide
information on both the magnitude (size) of a force and the direction it is acting.

20N 20N 20N 20N 20N 20N 20N 20N 100N 20N 20N 20N

2 0N 65N 65N
N 20N
20
20N
20N
20N
20N With the force-arrows drawn on, predict the
outcome of each Tug o’ War…
 Vectors Challenge & Develop

As you know from year 7, the forces acting on an object can be combined to identify
the single resultant force.

becomes; 20 20 20 20 20 20 20 20

20N 20N 20N 20N 20N 20N 20N 20N

which simplifies to;

80 N 80 N
As these two forces are equal and opposite,
there is no resultant force; or rather,
the resultant force is zero newtons (0 N)

This type of sum can be done for any vector quantity.

 Vectors Challenge & Develop

Calculate the resultant forces in these examples;

1. 4N
2.
6N
10N 3N 4N
1N

5.
3. 3N 4. 4N
6N
4N

4N
7N There4Nis no
2N resultant force in
this case
 Vectors Challenge & Develop

What you have just done, is a vector sum!

The sum of vectors with the same direction The sum of vectors with the opposite
requires adding their magnitudes. directions requires subtracting their
magnitudes
Vectors Challenge & Develop

Practice your vector addition skills using

the worksheet
Vectors Challenge & Develop
Vectors Challenge & Develop
 Vectors Explain

Force is a vector. The vector sum of all the forces on an object gives a net or
resultant force.
Look at the forces on this object: To see the vector sum of the forces, add the
vectors together nose-to-tail.
4N
3N

4N 4N

3N 3N 3N

There is no ‘net’ or resultant force on the

mass: the forces are balanced.

4N
 Vectors Explain

The forces on this mass are

unbalanced:
A vector sum can still be used to calculate the size of the
resultant force.
12 N Draw your arrows nose-to-tail;

12 N 6N

6N

6N
6N So the resultant force is
12 N – 6N = 6N upwards.

What would you do if the arrows were not parallel

to each other?
 Vectors Explain

The forces on this mass are also unbalanced:

What is the resultant force?
8N
Add the vectors together again, nose-to-tail.
3N

4N

3N 8N

This is the resultant force.

4N

How could you calculate the size and direction of this force?
 Vectors Explain

There are two options to calculate the size of the resultant force.
1. Draw a scale diagram
8N
2. Use trigonometry

● Draw each force arrow nose-to-tail;

3N
● Draw the resultant force arrow;
● Use a ruler & protractor to measure the
length (magnitude) and angle.
4N
This resultant force is
5 N at 37o from the horizontal

(If you are using squared paper, the scale can be as easy as 1 square = 1 N. Alternatively you
could use a ruler, with 1 cm = 1 N.)
 Vectors Explain

For completeness, the trigonometry method uses Pythagoras’ Theorem.

● Draw each arrow nose-to-tail;
● Draw the resultant force arrow;
● Make the resultant force arrow the hypotenuse of a triangle.
● Use a2 + b2 = c2 and soh cah toa to calculate the magnitude of the arrow and
it’s angle from the vertical.

3N 3N
F = √42 + 32 = 5 N
4N 4N θ
θ = tan-1(3/4) = 37°
8N F
So the resultant force is
5 N at 37o from the vertical

You meet this method in GCSE Maths! Don’t worry about it for now!
Vectors Challenge & Develop

To calculate a resultant force using vectors and a scale diagram;

● Choose a suitable scale
● Draw each force arrow nose-to-tail;
● Draw the resultant force arrow;
● Choose a direction to be positive
(usually the direction of the largest force)
● Use a ruler & protractor to measure the length (magnitude) and angle.
 Vectors Explain

What is the resultant force on this satellite?

● Draw each arrow nose-to-tail;
● Draw the resultant force arrow;
● Use a ruler & protractor to measure the length
(magnitude) and angle.
8N

I am using a scale of 1
square = 4 N
20 N
So the resultant force is
21.5 N at 22o
from the vertical
+ve
(Remember to choose a direction to be positive – in this case, the direction of the largest
force seems sensible!)
 Vectors Explain

10 N
What is the resultant force on this Seagull?
On your squared paper;
● Draw each arrow nose-to-tail;
● Draw the resultant force arrow;
● Choose a direction to be positive; 6N
● Use a ruler & protractor to measure the length
2N
(magnitude) and angle.

+ve

So the resultant force is

10 N at 37o from the vertical

(It’s flying up and away from us as we look)

1 Square = 2 N
 Vectors Explain

What are the resultant forces in these cases?

250 N

200 N
400 N
210 N 90 N 53o 200 N
90o
150 N
NB: the green arrows are at 90o
to each other
On your squared paper;
● Draw each force arrow nose-to-tail;
● Draw the resultant force arrow;
● Choose a direction to be positive;
● Measure the length (magnitude) and angle. For the Train, try a scale of 1 Square = 25 N
 Vectors Explain

200 N
Resultant Force
290 N at 44o from
the vertical
210 N

On your squared paper;

● Draw each force arrow nose-to-tail;
● Draw the resultant force arrow;
● Choose a direction to be positive;
● Measure the length (magnitude) and angle. For the Train, try a scale of 1 Square = 25 N
 Vectors Explain

400 N
90 N

Resultant Force
410 N at 13o below the horizontal

On your squared paper;

● Draw each force arrow nose-to-tail;
● Draw the resultant force arrow;
● Choose a direction to be positive;
● Measure the length (magnitude) and angle. For the Boat, I used a scale of 1 Square = 50 N
 Vectors Explain

For the Balloon, I used a scale of 1 cm = 50N

250 N
& drew the green arrows first!

53o 200 N Resultant Force

90o 0N
150 N
NB: the green arrows are at 90o
to each other

On your squared paper;

● Draw each force arrow nose-to-tail;
● Draw the resultant force arrow;
● Choose a direction to be positive;
● Measure the length (magnitude) and angle.
Vectors Explain

Now work through the attached sheet to

calculate the resultant force in each case.
 Vectors Explain
Measuring all angles from the top of the +ve
page! (i.e. 0O is straight up)
Force A: 3 N Right
Force B: 4 N Up A
F
Force C: 5 N Down Resultant
Force D: 2 N Right B 5.4 N at 112o
E
Resultant C Resultant
5 N at 143o 2.8 N at 315o
Foce E: 2 N Up
Force F: 2 N Left D J
Resultant
Force G: 2 N Down Resultant 4.5 N at 233o
Force H: 10 N Right 10.2 N at 101o K
G
Force J: 3 N Right H L
Force K: 4N Down
Force L: 5 N Left
 Vectors Consolidate & Apply

When this process is used in reverse it is called ‘resolving’.

For example; Resolve this force into horizontal & vertical components

Since it’s already drawn to scale,

we just ‘complete the triangle’
and measure the lengths of the
sides. 130 N

Horizontal Component;
120mm = 120 N
Vertical Component;
60mm = 60 N

20mm = 20 N
Vectors Consolidate & Apply

Resolve these forces into horizontal & vertical components

1) 2) 3)

3N
4N 2N
6N 4N
4N

4) 5) 6)
1.5 N
6N 6N 4N

3N 5N 4N
 Vectors Consolidate & Apply

We can also resolve Vectors by drawing a diagram to scale first.

For example;
This horse is pulling the cart with a force of 500 N.

N
500
37o

The shaft which connects the cart to the horse is at an angle

of 37o. This means some of the force is pulling the cart up,
whilst the rest pulls the cart forward.

How much force is pulling the cart forwards?

 Vectors Consolidate & Apply

Our question has presented us with the hypotenuse of a

N
right-angled triangle. 500
37o
Therefore, we draw a scale diagram;
1. With a suitable scale (e.g. 1 square = 100 N) draw a
line representing the 500 N at 37o
2. Then add a horizontal arrow ending directly below
the end of the first line
3. & a vertical arrow between ends of the two lines
4. Double check your arrows correctly show the
direction in which each force acts
5. Finally measure the lengths of the horizontal and
vertical lines. Use the scale for the first line to
convert these lengths to the corresponding forces. Horizontal Force =
400 N
Vectors Consolidate & Apply

Working backwards from a resultant force to the individual components is called

‘resolving vectors’ (or in this exact instance, resolving forces!)

The Steps are;

1. Draw a line representing the force at the correct angle
2. Draw a horizontal line ending directly below the end of the first line
3. Draw a vertical line between ends of the two lines
4. Check the arrow heads show the direction in which each force acts
5. Measure the lengths of the horizontal and vertical lines.
6. Use the scale for the first line to convert these lengths to the corresponding
forces.

(In some questions you might only need to measure either the horizontal or vertical force, rather than both)
 Vectors Consolidate & Apply

Here’s another example;

A force of magnitude 100 N acts in a direction of 40° to the horizontal. Resolve this force
into horizontal and vertical components.
Horizontal component = 77 mm = 77 N

Vertical Component = 63 mm = 63 N
Vectors Consolidate & Apply

Work out the vertical component of the force shown

below;

The diagram shows the small aircraft being used to tow a glider.
The tension in the cable is 2000 N
The cable makes an angle of 20° with the horizontal.
 Vectors Consolidate & Apply

The diagram shows the small aircraft being used to

tow a glider. The tension in the cable is 2000 N. The
cable makes an angle of 20° with the horizontal.

Vertical component = 33mm = 660 N

Scale: 10mm = 200 N

 Vectors Consolidate & Apply

Now try these two questions;

0 N 17
25 0N WIND
40o 20o
(not drawn to scale)
60N

The resultant force on a hot air balloon

A Dog’s owner pulls on a lead with a force of
causes it to rise with a force of 170 N at a
250 N at an angle of 40o above the horizontal.
angle of 20o.
What is the size of the vertical force?
If the Burner supplies a lift of 60 N,
what is the contribution of the wind? (the
size of the horizontal force)
 Vectors Consolidate & Apply

0 N
25
40o

A Dog’s owner pulls on a lead with a force of

250 N at an angle of 40o above the horizontal.
What is the size of the vertical force?
 Vectors Consolidate & Apply

17
0N WIND
20o Horizontal Force (the Wind)
60N = 80mm = 160 N
(not drawn to scale)

85m
m = 17
0N
The resultant force on a hot air balloon
causes it to rise with a force of 170 N at a
angle of 20o.
If the Burner supplies a lift of 60 N,
what is the contribution of the wind? (the
size of the horizontal force) Scale: 10mm = 20 N
Vectors Consolidate & Apply

Now try these questions;

1. Two forces act on an object. One force is 5N to the right and the other force
is 12N upwards. Draw a scale diagram to find the magnitude and direction of
the resultant force.
2. A 12N force is pulling an object to the right and a 16N force is pulling it
downwards. A third force is needed to keep the object still, so that there is no
resultant force. Determine the size of the third force by drawing a scale
diagram.
3. A 14.1N force acts at 45° above the horizontal to the right. Determine the
vertical and horizontal components of this force.
4. The 14.1N force in question 3 acts together with a 14.1N force which points
at 45° below the horizontal.
Determine the resultant force produced by these two forces.
 Vectors Consolidate & Apply

1. Two forces act on an object. One force is 5N to the right and

the other force is 12N upwards.
Draw a scale diagram to find the magnitude and direction of
the resultant force.

12 N
13 N

67o

5N
 Vectors Consolidate & Apply

2. A 12N force is pulling an object to the right and a 16N force

is pulling it downwards. A third force is needed to keep the 12 N
object still, so that there is no resultant force. Determine the
size of the third force by drawing a scale diagram.

125mm = 12 N
25 N

what it looks like

drawn ‘normally’
 Vectors Consolidate & Apply

3. A 14.1N force acts at 45° above the horizontal to the right.

Determine the vertical and horizontal components of this
force.

14.1 N
10 N

45o

10 N
 Vectors Consolidate & Apply

4. The 14.1N force in question 3 acts

together with a 14.1N force which points
at 45° below the horizontal.
Determine the resultant force produced 14.1 N 10 N
by these two forces. 45o
10 N
Resultant
45o force = 20 N right

14.1 N 10 N