Vectors and Scalars

Understanding quantities with and without direction - the foundation of mechanics

Grade 10 Physical Sciences

Key Concept: Scalars have magnitude only, while vectors have both magnitude and direction. This distinction is crucial for accurately describing physical quantities.

1. Definitions

Scalars

Definition: Quantities with magnitude only

Common Scalars

Mass (m): kilogram (kg)

Time (t): second (s)

Distance (d): meter (m)

Speed (s): m/s

Energy (E): joule (J)

Temperature (T): °C or K

Vectors

Definition: Quantities with both magnitude and direction

Common Vectors

Displacement (s): meter (m) + direction

Velocity (v): m/s + direction

Acceleration (a): m/s² + direction

Force (F): newton (N) + direction

2. Vector Representation

Graphical Representation

Vectors are represented as arrows where:

Length = magnitude (drawn to scale)
Arrowhead = direction

Vector Builder

Adjust the magnitude and direction to see how the vector changes

Magnitude: 150

Direction: 0°

Vector Notation

Vectors are denoted with an arrow above the symbol: \(\vec{v}\) or \(\vec{F}\)

3. Direction Conventions

Compass Points

45°

Example: 30° North of East

Bearing

Angles measured clockwise from North

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

Bearing 045° = Northeast

Relative Direction

Directions can also be described using positive/negative signs (e.g., +x, -x, up, down, left, right).

4. Resultant Vectors

The resultant vector is a single vector that has the same effect as all original vectors acting together.

Vectors in the Same Direction

\(\vec{R} = \vec{A} + \vec{B}\)

Vectors in Opposite Directions

\(\vec{R} = \vec{A} - \vec{B}\) (Right - Left)

Tail-to-Head Method

Place the tail of the second vector at the head of the first. The resultant goes from the start of the first to the end of the last.

Vector A

Vector B

Resultant R

Resultant Vector Calculator

For vectors in the same line:

Vector A (units):

Vector B (units):

Resultant: 15 units to the right

5. Distance vs Displacement

Quantity	Distance	Displacement
Type	Scalar	Vector
Definition	Total path length traveled	Change in position (straight line from start to end)
Direction	No direction	Has direction
Sign	Always positive	Can be positive, negative, or zero

Example: A person walks 3 m east, then 4 m west.

Distance: 3 + 4 = 7 m

Displacement: 3 - 4 = -1 m (1 m west)

A person walks 5 km north, then 3 km south. What are the distance and displacement?

Match the Quantity

Distance

Displacement

Velocity

Speed

Scalar: total path length

Vector: change in position

Vector: rate of change of displacement

Scalar: rate of change of distance

Complete the Sentences

Scalars have only .

Vectors have both magnitude and .

Distance is a quantity.

Displacement is a quantity.

Test Your Understanding

1. Which of the following is a vector quantity?

2. A car travels 10 km east, then 6 km west. The displacement is:

3. Which method is used to find the resultant vector graphically?

4. A bearing of 090° corresponds to which direction?

Key Terms

Scalar Vector Magnitude Direction Displacement Distance Resultant Tail-to-head Bearing Compass points Velocity Force Acceleration Speed

Key Takeaways

Scalars have magnitude only (e.g., mass, time, distance, speed)
Vectors have both magnitude and direction (e.g., displacement, velocity, force)
Vectors are represented by arrows - length = magnitude, arrowhead = direction
Direction can be given as compass points, bearings, or +/- signs
Resultant vector = single vector with same effect as all vectors combined
Same direction: add vectors; Opposite direction: subtract vectors
Tail-to-head method: place vectors head-to-tail, resultant from start to finish
Distance (scalar) = total path length; Displacement (vector) = change in position

Mechanics Motion in 1D