1.2 Scalars & Vectors
What are Physical Quantities?
In physics, every measurement is a quantity. We can split all quantities into two groups: scalars and vectors.
In physics, every measurement is a quantity. We can split all quantities into two groups: scalars and vectors.
⚡ Key Concept:
• A scalar is a quantity that only has a magnitude (size)
• A vector is a quantity that has both a magnitude AND a direction
• A vector is a quantity that has both a magnitude AND a direction
Scalar
Magnitude Only
Example: 5 m/s
"How fast?"
Vector
Magnitude + Direction
Example: 5 m/s East
"How fast AND which way?"
Real life scenario:
If someone asks "How far is the shop?" — you give a scalar (e.g., "500 metres")
If someone asks "How do I get to the shop?" — you need a vector (e.g., "500 metres North")
If someone asks "How do I get to the shop?" — you need a vector (e.g., "500 metres North")
Understanding the Difference
The easiest way to understand scalars and vectors is to compare related quantities.
• Speed
• Mass
• Time
• Energy
• Temperature
• Velocity
• Acceleration
• Force
• Momentum
• Weight
The easiest way to understand scalars and vectors is to compare related quantities.
⚡ Speed vs Velocity:
• Speed (scalar): How fast you're moving — just a number
• Velocity (vector): How fast AND in which direction
• Velocity (vector): How fast AND in which direction
Example: Running at 5 m/s
Speed: 5 m/s (scalar — no direction)
Velocity:
• 5 m/s East → written as +5 m/s
• 5 m/s West → written as −5 m/s
The speed is the same (5 m/s), but the velocities are opposite.
Velocity:
• 5 m/s East → written as +5 m/s
• 5 m/s West → written as −5 m/s
→ +5 m/s (East)
← −5 m/s (West)
Distance vs Displacement
Distance (Scalar)
Total length travelled
"I walked 100 m"
Total length travelled
"I walked 100 m"
Displacement (Vector)
Straight-line distance + direction
"I'm 60 m North of where I started"
Straight-line distance + direction
"I'm 60 m North of where I started"
Speed vs Velocity
Speed (Scalar)
How fast you're moving
"The car is going 30 m/s"
How fast you're moving
"The car is going 30 m/s"
Velocity (Vector)
Speed + direction
"The car is going 30 m/s North"
Speed + direction
"The car is going 30 m/s North"
Common Scalars
• Distance• Speed
• Mass
• Time
• Energy
• Temperature
Common Vectors
• Displacement• Velocity
• Acceleration
• Force
• Momentum
• Weight
🎯 Scalar or Vector Practice
How We Show Direction
Because vectors have direction, we need ways to represent this. There are several methods:
Because vectors have direction, we need ways to represent this. There are several methods:
⚡ Methods to Show Direction:
• Compass directions: North, South, East, West
• Positive/Negative signs: +5 m/s (right), −5 m/s (left)
• Angles: 30° from horizontal
• Arrows: Length = magnitude, direction = arrow points
• Positive/Negative signs: +5 m/s (right), −5 m/s (left)
• Angles: 30° from horizontal
• Arrows: Length = magnitude, direction = arrow points
Example: Using Signs for Direction
We often choose a direction to be "positive".
If East = positive (+):
• 5 m/s East → +5 m/s
• 5 m/s West → −5 m/s
If Up = positive (+):
• 10 N upward → +10 N
• 10 N downward → −10 N
If East = positive (+):
• 5 m/s East → +5 m/s
• 5 m/s West → −5 m/s
If Up = positive (+):
• 10 N upward → +10 N
• 10 N downward → −10 N
Example: Four Objects Moving at Same Speed
All balls below are moving at 5 m/s. Their speeds are identical, but their velocities are different because they're going in different directions.
Speed: All = 5 m/s ✓
Velocity: All different! (5 m/s E, 5 m/s W, 5 m/s N, 5 m/s S)
5
5
5
5
Speed: All = 5 m/s ✓
Velocity: All different! (5 m/s E, 5 m/s W, 5 m/s N, 5 m/s S)
🧮 Vector Builder:
Create a vector by choosing a magnitude and direction:
Vector: 10 m/s East (+10 m/s)
🎯 Direction Practice
The Importance of Direction in Physics
Direction isn't just extra information, it tells you how to carry out a calculation.
Direction isn't just extra information, it tells you how to carry out a calculation.
⚡ Remember:
When vectors point in the same direction → they ADD
When vectors point in opposite directions → they SUBTRACT
This is why direction matters for calculations.
When vectors point in opposite directions → they SUBTRACT
This is why direction matters for calculations.
Example 1: Two Forces in Same Direction
Two people push a box to the right:
• Person A: 20 N right (+20 N)
• Person B: 15 N right (+15 N)
Total force: +20 + (+15) = +35 N right
• Person A: 20 N right (+20 N)
• Person B: 15 N right (+15 N)
Total force: +20 + (+15) = +35 N right
20 N →
+
15 N →
=
35 N →
Example 2: Two Forces in Opposite Directions
Two people push a box in opposite directions:
• Person A: 20 N right (+20 N)
• Person B: 15 N left (−15 N)
Total force: +20 + (−15) = +5 N right
• Person A: 20 N right (+20 N)
• Person B: 15 N left (−15 N)
Total force: +20 + (−15) = +5 N right
20 N →
+
← 15 N
=
5 N →
Example 3: Distance vs Displacement
You walk 40 m East, then 30 m West.
Distance (scalar): 40 + 30 = 70 m (total path)
Displacement (vector): +40 + (−30) = +10 m East (final position)
Distance (scalar): 40 + 30 = 70 m (total path)
Displacement (vector): +40 + (−30) = +10 m East (final position)
40 m →
then
← 30 m
=
10 m → from start
🎯 Vector Addition Practice
Quick Reference Table
| Quantity | Type | Has Direction? | Example |
|---|---|---|---|
| Distance | Scalar | No | 50 m |
| Displacement | Vector | Yes | 50 m North |
| Speed | Scalar | No | 20 m/s |
| Velocity | Vector | Yes | 20 m/s East |
| Mass | Scalar | No | 5 kg |
| Weight | Vector | Yes | 50 N downward |
| Time | Scalar | No | 10 s |
| Acceleration | Vector | Yes | 9.8 m/s² down |
| Energy | Scalar | No | 100 J |
| Force | Vector | Yes | 10 N right |
💡 Memory:
Scalars are Simple — just a Size
Vectors have Very important direction
Vectors have Very important direction
Real Life Uses:
• Navigation: GPS uses vectors (displacement) not just distance
• Sports: A footballer's velocity determines where they'll intercept the ball
• Weather: Wind is described as a vector (10 mph NE, not just "10 mph")
• Aviation: Pilots must account for wind vectors when flying
• Engineering: Forces on bridges must be calculated with direction
• Navigation: GPS uses vectors (displacement) not just distance
• Sports: A footballer's velocity determines where they'll intercept the ball
• Weather: Wind is described as a vector (10 mph NE, not just "10 mph")
• Aviation: Pilots must account for wind vectors when flying
• Engineering: Forces on bridges must be calculated with direction