1.4 Scalars and Vectors
Objectives:
•
•
•
Understand the difference between scalar and vector quantities and give examples of scalar
and vector quantities included in the syllabus
Add and subtract coplanar vectors
Represent a vector as two perpendicular components
Scalar
Vector
Definition
Physical quantities that have
magnitude only.
Physical Quantities that have both
magnitude and direction.
Examples
Distance
Speed
Mass
Energy
Time
Work
Power
Displacement
Velocity
Acceleration
Force
Momentum
Weight
Representation
A scalar is represented by its
magnitude (numeric value) and unit
only.
A vector is represented by its direction
such as North, west, east, south or
degrees or with arrows along with its
magnitude and unit.
Representing Vectors:
Length of the arrow, drawn to scale represent magnitude, while where the arrow is pointing with
respect to the reference represent direction.
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Parallel Vectors:
Vector Addition (Head to tail rule) – Graphical Method:
(Head to tail rule)
• Choose a suitable scale. Your diagrams should be
reasonably large (covering the area given).
• Draw a line to represent the first vector.
• North at the top of the page.
You will connect the head of one vector to the tail of the another.
Vectors can be moved
parallel to themselves
without changing the
resultant.
The Red Arrow
represents the
resultant of the two
vectors.
Vector Subtraction – Graphical Method:
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Coplanar Vectors:
When 3 or more vectors need to be added, the same principles apply, provided the vectors are all on the
same plane i.e., coplanar
To subtract 2 vectors, reverse the direction i.e., change the sign of the vector to be subtracted, and add.
⃗𝑨 + ⃗𝑩
⃗ + ⃗𝑪 = ⃗𝑹
⃗
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Mathematical Methods:
If the two vectors are at right angle to each other, we can use Pythagoras theorem to solve it
To find magnitude
𝑪𝟐 = 𝑨𝟐 + 𝑩𝟐
To find direction
tan 𝜃 =
𝒐𝒑𝒑
𝒂𝒅𝒋
Resolution of Vectors:
We know when two vectors are added together, it can be represented by a single resultant vector that
has the total/same effect of the two vectors.
Now we are going to be resolving a single vector into two components.
“if two perpendicular vectors have a resultant vector, that resultant vector can also be resolved into its
two perpendicular vectors”
We resolve a vector into two components
Component along X-axis called x-component - See the green arrows to know how to represent them.
Component along Y-axis called Y-component
Which are perpendicular to each other. These components are called rectangular components of vector.
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Addition of vector by rectangular components method:
Consider two vectors acting in different directions.
1. Resolve both vectors into two rectangular components.
𝒗𝟏𝑿 = 𝒗𝟏 𝐜𝐨𝐬 𝜽𝟏
And
𝒗𝟏𝒀 = 𝒗𝟏 𝐬𝐢𝐧 𝜽𝟏
2. Resolve the second vector into its components.
𝒗𝟐𝑿 = 𝒗𝟐 𝐜𝐨𝐬 𝜽𝟐
And
𝒗𝟐𝒀 = 𝒗𝟐 𝐬𝐢𝐧 𝜽𝟐
Consider two vectors acting in different directions.
3. Find the magnitude of the x-component and ycomponent of the resultant.
𝑹𝒙 = 𝒗𝟏𝑿 + 𝒗𝟐𝑿
And
𝑹𝒚 = 𝒗𝟏𝒀 + 𝒗𝟐𝒀
4. Find the total magnitude of the resultant vector.
𝑹 = √𝑹𝟐𝑿 + 𝑹𝟐𝒀
5.
6. Find the angle of the resultant vector.
𝑹𝒀
𝜽 = 𝐭𝐚𝐧−𝟏
𝑹𝑿
Adding coplanar vectors by determining perpendicular components:
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The total force on the object in the xcomponents of 𝑭𝟏 and 𝑭𝟐
𝑭𝑿 = 𝑭𝟏 𝐜𝐨𝐬 𝜽𝟏 + 𝑭𝟐 𝒄𝒐𝒔 𝜽𝟐
The total force on the object in the ycomponents of 𝑭𝟏 and 𝑭𝟐
𝑭𝒚 = 𝑭𝟏 𝐬𝐢𝐧 𝜽𝟏 + (−𝑭𝟐 ) 𝐬𝐢𝐧 𝜽𝟐
Negative sign to indicate opposite directions.
Condition for Equilibrium:
Coplanar vectors can be represented by vector triangles.
In equilibrium, these are closed vector triangles. The
vectors, when joined together, form a closed path
If three forces acting on an object are in equilibrium; they form a closed triangle.
Summary:
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