Vectors and Scalars
Copyright © John O’Connor
St. Farnan’s PPS
Prosperous
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
A scalar quantity is a quantity that has
magnitude only and has no direction in space
Examples of Scalar Quantities:
 Length
 Area
 Volume
 Time
 Mass

A vector quantity is a quantity that has both
magnitude and a direction in space
Examples of Vector Quantities:
 Displacement
 Velocity
 Acceleration
 Force



Vector diagrams are
shown using an
arrow
The length of the
arrow represents its
magnitude
The direction of the
arrow shows its
direction

The resultant is the sum or the combined effect of
two vector quantities
Vectors in the same direction:
6N
4N
=
10 N
=
10 m
6m
4m
Vectors in opposite directions:
6 m s-1
10 m s-1
=
4 m s-1
6N
10 N
=
4N
When two vectors are joined
tail to tail

Complete the parallelogram

The resultant is found by
drawing the diagonal



When two vectors are joined
head to tail
Draw the resultant vector by
completing the triangle
2004 HL Section B Q5 (a)
Two forces are applied to a body, as shown. What is the magnitude
and direction of the resultant force acting on the body?
Solution:


Complete the parallelogram (rectangle)
The diagonal of the parallelogram ac
represents the resultant force
The magnitude of the resultant is found using
Pythagoras’ Theorem on the triangle abc
a
Magnitude  ac  12  5
ac  13 N
2
12
Direction of ac : tan  
5
12
   tan 1  67
5
2
b

12 N
d
θ
5N

5
12
c
Resultant displacement is 13 N 67º
with the 5 N force
Find the magnitude (correct to two decimal places) and direction of the
resultant of the three forces shown below.
Solution:
Find the resultant of the two 5 N forces first (do right angles first)
ac  52  52  50  7.07 N
5
tan    1    45
5



Now find the resultant of the 10 N and
7.07 N forces
The 2 forces are in a straight line (45º +
135º = 180º) and in opposite directions
So, Resultant = 10 N – 7.07 N = 2.93 N
in the direction of the 10 N force
5
d
c
5
5N

a
90º
θ
45º
135º
5N
b








What is a scalar quantity?
Give 2 examples
What is a vector quantity?
Give 2 examples
How are vectors represented?
What is the resultant of 2 vector quantities?
What is the triangle law?
What is the parallelogram law?


When resolving a vector into
components we are doing the
opposite to finding the resultant
We usually resolve a vector into
components that are
perpendicular to each other
Here a vector v is resolved into
an x component and a y
component
y

x


Here we see a table
being pulled by a force
of 50 N at a 30º angle
to the horizontal
When resolved we see
that this is the same as
pulling the table up
with a force of 25 N
and pulling it
horizontally with a
force of 43.3 N
y=25 N
30º
x=43.3 N

We can see that it
would be more
efficient to pull the
table with a
horizontal force of
50 N

If a vector of magnitude v and makes an angle θ
with the horizontal then the magnitude of the
components are:
y=v Sin θ
x = v Cos θ
y
θ
y = v Sin θ

Proof:


x
Cos 
v
x  vCos
x=vx Cos θ
y
Sin  
v
y  vSin
2002 HL Sample Paper Section B Q5 (a)
A force of 15 N acts on a box as shown. What is the horizontal
component of the force?
Horizontal Component  x  15Cos60  7.5 N
Vertical Component  y  15Sin 60  12.99 N
Vertical
12.99 N
Component
Solution:
60º
Horizontal
7.5 N
Component
2003 HL Section B Q6

A person in a wheelchair is moving up a ramp at constant speed. Their
total weight is 900 N. The ramp makes an angle of 10º with the
horizontal. Calculate the force required to keep the wheelchair moving
at constant speed up the ramp. (You may ignore the effects of friction).
Solution:
If the wheelchair is moving at constant speed (no acceleration), then
the force that moves it up the ramp must be the same as the
component of it’s weight parallel to the ramp.
Complete the parallelogram.
Component of weight
10º
parallel to ramp:
 900Sin10  156.28 N
80º
10º
Component of weight
perpendicular to ramp:
 900Cos10  886.33 N
900 N
886.33 N



If a vector of magnitude v has two
perpendicular components x and y, and v
makes and angle θ with the x component
then the magnitude of the components are:
x= v Cos θ
y= v Sin θ
y=v Sin θ
y
θ
x=v Cosθ