Physics and Physical
Measurement
Topic 1.3 Scalars and Vectors
Scalars Quantities
 Scalars
can be completely described by
magnitude (size)
 Scalars can be added algebraically
 They are expressed as positive or negative
numbers and a unit
 examples include :- mass, electric charge,
distance, speed, energy
Vector Quantities

Vectors need both a magnitude and a direction
to describe them (also a point of application)

They need to be added, subtracted and multiplied
in a special way
 Examples
:- velocity, weight, acceleration,
displacement, momentum, force
Addition and Subtraction
 The
Resultant (Net) is the result vector that
comes from adding or subtracting a multiple
vectors
 If vectors have the same or opposite
directions the vector addition can be done
simply:
• same direction : add
• opposite direction : subtract
Co-planar vectors

The addition of co-planar vectors that do not have
the same or opposite direction can be solved by
using either…
• Scaled Drawing
– Vectors can be represented by a straight line
segment with an arrow at the end
• Pythagoras’ theorem and trigonometry
Scaled Drawing method triangle:
Choose an appropriate scale to fit the space
 Draw one vector represented by straight arrow
equal to the scaled length and with the arrow head
pointing in the proper direction
 Start drawing the second vector at the tip of the
first
 The resultant vector is the third side of a triangle
and the arrow head points in the direction from the
‘free’ tail to the ‘free’ tip

Example
R=a+b
a
+
b
=
Measure the length and direction of the
resultant in the drawing. Use your scale to
convert to real life value.
Scaled Drawing - Parallelogram
 Draw
the two vectors tail to tail, to scale
and with the correct directions
 Then complete the parallelogram
 The diagonal starting where the two tails
meet and finishing where the two arrows
meet becomes the resultant vector
Example
R=a+b
a
+
b
=
More than 2 vectors
 If
there are more than 2 co-planar vectors to
be added, place them all head to tail to form
polygon
 The resultant is drawn from the ‘free’ tail at
the beginning to the ‘free’ tip at the end.
 The
order doesn’t matter!
Subtraction of Vectors
 To
subtract a vector, you reverse the
direction of that vector to get the negative
of it, then you simply add that vectors
Example
a
-
b
=
R = a + (- b)
-b
Multiplying Vectors
A vector multiplied by a scalar gives a vector with
the same direction as the vector and magnitude
equal to the product of the scalar and a vector
magnitude
 A vector divided by a scalar gives a vector with
same direction as the vector and magnitude equal
to the vector magnitude divided by the scalar
 You don’t need to be able to multiply a vector by
another vector

Resolving Vectors
 The
process of finding the Components of
vectors is called Resolving vectors
 Just
as 2 vectors can be added to give a
resultant, a single vector can be split into 2
components or parts
The Rule
 Any
vector can be split into two
perpendicular components
 These could be the vertical and horizontal
components
Vertical component
Horizontal component
 Or
these could be parallel to and
perpendicular to an inclined plane
Doing the Trigonometry
V
Sin  = opp/hyp = y/V
y

x
V sin 

Therefore y = Vsin 
Which is the vertical
component
Cos  = adj/hyp = x/V
Therefore x = Vcos 
Which is the horizontal
component
V cos 
Adding 2 or More Vectors by
Components
 First
resolve into components (making sure
that all are in the same 2 directions)
 Then add the all the components in each of
the 2 directions
 Recombine them into a resultant vector
using Pythagoras´ theorem