This document discusses scalars and vectors. It defines a scalar as having only magnitude, and provides examples like length, area and time. A vector is defined as having both magnitude and direction, and examples include displacement, velocity and force. Vectors are represented by arrows, with length indicating magnitude and direction. The resultant of vectors is found by combining them using the triangle law or parallelogram law. Resolving a vector breaks it into perpendicular components using trigonometry.

1. Vectors
and
Scalars

2.  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

3.  A vector quantity is a quantity that has both
magnitude and a direction in space
Examples of Vector Quantities:
 Displacement
 Velocity
 Acceleration
 Force

4.  Vector diagrams are
shown using an
arrow
 The length of the
arrow represents its
magnitude
 The direction of the
arrow shows its
direction

5. Vectors in opposite directions:
6 m s-1 10 m s-1 = 4 m s-1
6 N 10 N = 4 N
Vectors in the same direction:
6 N 4 N = 10 N
6 m
= 10 m
4 m
 The resultant is the sum or the combined effect of
two vector quantities

6.  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

7. Solution:
 Complete the parallelogram (rectangle)
θ
 The diagonal of the parallelogram ac
represents the resultant force
Two forces are applied to a body, as shown. What is the magnitude
and direction of the resultant force acting on the body?
5N
12 N
5
12
a
b c
d
 The magnitude of the resultant is found using
Pythagoras’ Theorem on the triangle abc
N13
512Magnitude 22


ac
ac



67
5
12
tan
5
12
tan:ofDirection
1

ac
 Resultant displacement is 13 N 67º
with the 5 N force

8. 45º
5 N
90ºθ
Find the magnitude (correct to two decimal places) and direction of the
resultant of the three forces shown below.
5N
5
5
Solution:
 Find the resultant of the two 5 N forces first (do right angles first)
a b
cd
N07.75055 22
ac
 451
5
5
tan 
135º
 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

9.  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?

10.  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
y
x
 Here a vector v is resolved into
an x component and a y
component

11.  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
x=43.3 N
30º
 We can see that it
would be more
efficient to pull the
table with a
horizontal force of
50 N

12.  If a vector of magnitude v and makes an angle θ
with the horizontal then the magnitude of the
components are:
 x = v Cos θ
 y = v Sin θ
y=v Sin θ
x=v Cos θ
θ
y
 Proof:
v
x
Cos 
vCosx 
v
y
Sin 
vSiny 
x

13. 60º
A force of 15 N acts on a box as shown. What is the horizontal
component of the force?
Vertical
Component
Horizontal
Component
Solution:
N5.76015ComponentHorizontal  Cosx
N99.126015ComponentVertical  Siny
7.5 N
12.99N

14.  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.
10º
10º80º
900 N
Complete the parallelogram.
Component of weight
parallel to ramp:
N28.15610900  Sin
Component of weight
perpendicular to ramp:
N33.88610900  Cos
886.33 N

15.  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
x=v Cosθ
θ
y