Scalar and Vector Quantities | JHS Mathematics

Introduction

In physics, we classify quantities as either scalar or vector based on whether they have only magnitude or both magnitude and direction.

\[ \text{Scalar} \neq \text{Vector} \]

                                
                                     Key Differences
                                
                                        Feature
                                        Scalar Quantity
                                        Vector Quantity
                                    
                                        Magnitude
                                        Yes
                                        Yes
                                    
                                        Direction
                                        No
                                        Yes
                                    
                                        Representation
                                        Number only
                                        Number + direction
                                    
                                        Examples
                                        Mass, time
                                        Force, velocity

Feature	Scalar Quantity	Vector Quantity
Magnitude	Yes	Yes
Direction	No	Yes
Representation	Number only	Number + direction
Examples	Mass, time	Force, velocity

Examples

Classifying Quantities

Let's classify these common physical quantities:

Scalar Quantities

Mass (kg)
Time (s)
Speed (m/s)
Distance (m)
Volume (m³)
Energy (J)
Work (J)

Vector Quantities

Force (N)
Velocity (m/s)
Weight (N)
Momentum (kg·m/s)
Displacement (m)
Acceleration (m/s²)

                                
                                     Important Notes
                                
                                Speed vs Velocity: Speed is scalar (how fast), velocity is vector (how fast + direction)
Distance vs Displacement: Distance is scalar (total path), displacement is vector (shortest path + direction)
Mass vs Weight: Mass is scalar, weight is a force (vector)

Vectors

Vector Representation

A vector can be represented as movement along a given bearing (direction). It has:

Magnitude: Length of the vector (how much)
Direction: Angle from North (where to)

Example: A vector of 5 units at 045° bearing

Vector diagram showing 5 units at 45 degrees from North

Drawing Vectors

To draw a vector given its length and bearing:

Draw North direction (vertical line)
Measure angle clockwise from North
Draw line at this angle with specified length
Add arrowhead at end to show direction

Example: Draw a 6 cm vector at 120° bearing

Draw vertical North line
Measure 120° clockwise
Draw 6 cm line at this angle
Add arrowhead

Magnitude & Bearing

Understanding Bearing

Bearing is a 3-digit clockwise angle from North (000° to 360°):

North: 000° or 360°
East: 090°
South: 180°
West: 270°

Magnitude Calculation

The magnitude of a vector is its length. For movement:

\[ \text{Magnitude} = \text{Distance traveled along the vector} \]

Example: If you walk 5 km Northeast, the magnitude is 5 km

Zero Vector

Definition

A zero vector is a special case with:

Magnitude: 0 (no length)
Direction: Undefined (no direction)

It's represented as a single point

Practical Examples

Situations where zero vector occurs:

Returning to starting point (displacement = 0)
Balanced forces (net force = 0)
Object at rest (velocity = 0)

Practice Exercise

Question 1

Classify these as scalar or vector: speed, force, time, displacement

Scalar: speed, time

Vector: force, displacement

Question 2

What is the difference between distance and displacement?

Distance is scalar (total path length), displacement is vector (shortest path with direction)

Question 3

Draw a vector of 4 cm at 225° bearing

1. Draw North line

2. Measure 225° clockwise (South-West direction)

3. Draw 4 cm line at this angle

4. Add arrowhead

Question 4

What is the bearing for East direction?

090°

Question 5

Give an example of a zero vector situation

Example: Walking 5 km North then 5 km South - your displacement is 0

Question 6

Why is velocity a vector quantity but speed is scalar?

Velocity includes both speed (magnitude) and direction, while speed only has magnitude

Ready for more challenges?

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Timed Test in 7 Levels

Test your skills with scalar and vector quantities