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Vectors and Scalars
Chapter 8
What is a Vector Quantity?
A quantity that has both Magnitude and a
Direction in space is called a Vector Quantity.
Examples of Vector Quantities
Displacement
Velocity
Momentum
Acceleration
Force
Electric Field Strength
Magnetic Flux Density
These are the ONLY Vector Quantities on your course.
All other quantities you meet are Scalar Quantities
What is a Scalar Quantity?
A quantity that has Magnitude only is called a
Scalar Quantity.
It has no direction in space.
Examples of Scalar Quantities
Distance
Work
Time
Power
Volume
Temperature
Pressure
Electric current
How do I represent a Vector quantity on a
diagram?
An amount of a vector quantity is
represented on a diagram by an
Arrow.
The Length of the arrow
represents its Magnitude.
The Direction in which the arrow
points shows its Direction.
What is the Resultant of two Vectors?
The Resultant of two Vectors is that single
vector that when acting alone has the same
effect as the other two vectors.
Finding the Resultant:
Vectors in same direction
The combined effect of a
displacement of 5 m East and a
displacement of 10 m East is a
displacement of 15 m East.
The combined effect of a force of 2
N and a force of 4 N acting on an
object in the same direction is a
force of 6 N acting on the object in
the same direction.
Finding the Resultant:
Vectors in the opposite direction
If the vectors are in opposite
directions:
The Magnitude of the Resultant
is found by subtracting the
magnitude of the smaller from the
magnitude of the bigger.
The Direction of the Resultant is
the direction of the bigger.
Finding the resultant of two non-collinear
vectors
If the vectors are not in the same straight line we
use the Parallelogram Law to find their resultant.
State the Parallelogram Law
If two vectors, drawn tail to tail, are the adjacent sides ab
and ad of the parallelogram abcd,
then the diagonal from a to c of this parallelogram is their
Resultant.
A ship moves with a constant velocity of 4 m s-1. A man walks across the
ship at right angles to the direction of motion of the ship with a velocity of
3 m s-1. Find the resultant velocity of the man in magnitude and direction.
Magnitude of resultant:
=
=
=  o with direction of motion of ship
length of arrow ac
42  32
=
Direction of resultant:
5 m s-1
(Using Pythagoras' Theorem)
Where tan  = 3/4

 =
36.87o
Experiment to Find the Resultant of two Forces
Use the equipment shown.
Adjust the size and direction of the
three forces until the knot in the
thread remains at rest.
Take the reading on each newton
balance, F
​ 1 ​, ​F2​ and F
​ 3​.
If we want the resultant of the two
forces ​F1​ ​and ​F​2​ its magnitude is
the reading on the third balance
(​F3​).
The direction of the resultant is in
the opposite direction to ​F​3​.
Resolving a Vector into Components
Resolving a Vector into
Components is expressing it
in terms of two other vectors
so that it is the Resultant of
these two.
These two are called
Components of the given
vector.
We need only study resolving a vector
into components that are at right angles
to each other. These are called
Perpendicular Components.
x is the Resultant of c and d.
c and d are two Perpendicular
Components of x.
A 100 N force acts on a cart in the direction shown. This force can be
resolved into a horizontal component ( x ) and a vertical component ( y ).
Each component represents the complete effect of the100 N in its
direction.
To prevent the cart from moving horizontally a force of 86.6 N acting to the
left is required.
If the weight of the cart is less than 50 N it will be lifted off the ground.
Calculating the Perpendicular Components
A vector of magnitude v has perpendicular components x and y.
v makes an angle  with the component x
The magnitudes of the components are:
x = v cos 
and
y = v sin 
Find the vertical and horizontal components of a vector
of magnitude 20 N acting at 60o to the horizontal.
The diagram shows the
components.
Horizontal component =
x = 20 cos 60o = 10 N
Vertical component =
y = 20 sin 60o = 17.32 N
A stone of weight 50 N rests on the sloped roof of a house. The roof is
inclined at 20o to the horizontal. Resolve the weight of the stone into
components parallel and perpendicular to the roof.
The diagram shows the weight vector and its 2 components.
Take care that you draw these in correctly.
Component of weight perpendicular
to roof
= 50 cos 20o = 46.98 N
Component of weight parallel to roof
= 50 sin 20o = 17.10 N