These are physical quantities that have both magnitude and direction.

Examples include:

1. Displacement
2. Velocity
3. Acceleration
4. Force
5. Pressure
6. Retardation
7. Momentum

A vector can be represented on paper by a directed line segment.

• Length of the line represents the magnitude of the vector.
• Arrowhead shows the direction of the vector.

1. Triangle method (head-to-tail method)

Steps:

1. Choose a suitable scale (e.g., 1 cm = 5 m).
2. Draw the first vector to scale.
3. From the head of the first vector, draw the second vector to scale.
4. Draw the resultant from the tail of the first vector to the head of the second vector.

Where:

1. Vi - First vector
2. V2 - Second vector
3. R - Resultant vector

Example 1

A man walks 20 m north and then 15 m east. Find his displacement from the starting point.

Solution

Demonstration

Let 1 cm = 5 m

• 20 m North = 4 cm
• 15 m East = 3 cm
• Draw to scale and complete triangle
• Resultant length = 5 cm → 25 m
• Angle from North ($\theta$) =

$\tan(\theta) = \frac{3}{4} \Rightarrow \theta = \tan^{-1}(0.75) = 36^\circ$

Answer: Displacement = 25 m at 36° East of North

2. Parallelogram method

• Draw two vectors from a common point.
• Complete the parallelogram using these two vectors as adjacent sides.
• The diagonal from the common point represents the resultant.

Parallelogram law of forces:

"If two vectors are represented by the sides of a parallelogram, the diagonal from their common starting point gives the resultant vector."

Example 2

Two forces, 40 N and 60 N, act on a body making an angle of 30° between them. Find the resultant force using the parallelogram method.

Solution

Demonstration

Choose a scale.

Let 1 cm = 10 N

40 N = 4 cm

60 N = 6 cm

Angle between them = 30°

Using scale drawing, diagonal = 9.7 cm → 97 N

Answer: Resultant force = 97 N

An equilibrant force is equal in magnitude but opposite in direction to the resultant force. It brings the object into equilibrium.

Example 3

A block is pulled by two forces:

• 4 N North
• 3 N North-East (45° to North)

Find the resultant force.

Solution

Use triangle method. Let 1 cm = 1 N

• Draw 4 cm vertically (North)
• From its head, draw 3 cm at 45°
• Measure resultant = 6.5 cm = 6.5 N

Answer: Resultant force = 6.5 N

Example 4

Two ropes, 3 m and 6 m long, are tied to the ceiling. A force of 100 N pulls their free ends. They make an angle of 30° between them. Find the tension in each rope.

Solution

Let 1 cm = 1 m

• 3 m = 3 cm
• 6 m = 6 cm
• Using scale drawing (parallelogram method), diagonal = 8.7 cm represents 100 N

• By using parallelogram method

Tension in 3 m rope:

$\frac{3}{8.7} \times 100 = 34.5 \, \text{N}$

Tension in 6 m rope:

$\frac{6}{8.7} \times 100 = 69.0 \, \text{N}$

Answer:

• Tension in 3 m rope = 34.5 N
• Tension in 6 m rope = 69.0 N