Vector & Scalar Quantities

Definition of Scalars

Scalars are physical quantities that have only magnitude (size or amount) but no direction. They are described by a single number and a unit.

Common examples of scalar quantities include:

• Speed (e.g. 30 m/s)
• Distance (e.g. 100 m)
• Mass (e.g. 5 kg)
• Time (e.g. 10 s)

Since scalars have no direction, calculations involving scalars use simple arithmetic (addition, subtraction, multiplication, division).

For example, if a runner completes a 400 m lap in 50 seconds, the speed is calculated as:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{400 \text{ m}}{50 \text{ s}} = 8 \text{ m/s}$

Definition of Vectors

Vectors are quantities that have both magnitude and direction. This means you must specify how much and which way.

Examples of vector quantities include:

• Velocity (e.g. 8 m/s north)
• Displacement (e.g. 50 m east)
• Force (e.g. 10 N downwards)
• Acceleration (e.g. 3 m/s² upwards)

Vectors are important in physics because direction affects how they combine and influence motion.

For instance, if a car moves 60 m east in 5 seconds, its velocity is:

$\text{Velocity} = \frac{\text{Displacement}}{\text{Time}} = \frac{60 \text{ m east}}{5 \text{ s}} = 12 \text{ m/s east}$

Representing Vectors

Vectors are commonly represented by arrows in diagrams:

• The length of the arrow shows the magnitude (how big the quantity is).
• The direction the arrow points shows the vector’s direction.

For example, a velocity vector pointing right with an arrow length representing 5 m/s shows an object moving at 5 m/s towards the right.

This visual representation helps when adding or subtracting vectors, or understanding motion in two dimensions.

Difference Between Scalars and Vectors

The key difference is direction:

• Scalars have magnitude only and no direction.
• Vectors have both magnitude and direction.

Because of this, calculations involving scalars use simple arithmetic, while vectors require vector addition or subtraction, considering both magnitude and direction.

For example, if you walk 3 km north and then 4 km east, your total distance (scalar) is 7 km, but your displacement (vector) is the straight-line distance from start to finish, which can be found using Pythagoras’ theorem:

$\text{Displacement} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \text{ km}$

The direction of this displacement is northeast, which can be calculated using trigonometry (tan θ = opposite/adjacent).