SCALAR AND VECTORS

Vectors and Scalars - Edwin Ogie Library

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Introduction to Vectors and Scalars

In Physics and Mathematics, quantities are categorized into two major types: scalars and vectors.

Scalar Quantities

Scalar quantities have only magnitude (size) but no direction. They are represented by a single value. Examples include:

• Distance
• Speed
• Mass
• Temperature
• Time

Vector Quantities

Vector quantities have both magnitude and direction. Examples include:

• Displacement
• Velocity
• Force
• Acceleration
• Momentum

How to Find Vectors

1. Using Scale Drawing Method:

In this method, vectors are drawn to scale on paper. The steps include choosing a scale, drawing the vectors to scale, and measuring them. Use a ruler and protractor to measure the lengths and angles.

2. Using Analytical Method:

For any vector \( \vec{V} \) with magnitude \( V \) and direction \( \theta \), the components are calculated as:

V_x = V cos(θ)

V_y = V sin(θ)

Parallelogram Law of Vectors

The Parallelogram Law states that if two vectors \( \vec{A} \) and \( \vec{B} \) are represented by two adjacent sides of a parallelogram, then the diagonal represents the resultant vector.

The formula for the resultant vector \( R \) is:

R = √(A² + B² + 2AB cos(θ))

Worked Example:

Given: \( A = 5 \, \text{N} \), \( B = 6 \, \text{N} \), \( \theta = 60^\circ \).

Find the resultant vector \( R \):

R = √(5² + 6² + 2 × 5 × 6 × cos(60°))

R = √(25 + 36 + 30) = √91 ≈ 9.53 N

Resolution of Vectors

Resolution of Vectors refers to breaking a vector into two perpendicular components, usually horizontal and vertical.

The components of a vector \( \vec{A} \) with magnitude \( A \) and angle \( θ \) are:

A_x = A cos(θ)

A_y = A sin(θ)

Worked Example:

Given: \( A = 10 \, \text{N} \), \( θ = 30^\circ \).

Find the horizontal and vertical components:

A_x = 10 × cos(30°) = 10 × 0.866 ≈ 8.66 N

A_y = 10 × sin(30°) = 10 × 0.5 = 5 N

JAMB Exam Practice Questions

1. Find the magnitude of the resultant vector if two vectors \( \vec{A} = 5 \, \text{N} \) and \( \vec{B} = 6 \, \text{N} \) make an angle of 60° with each other.

Answer:

R = √(A² + B² + 2AB cos(θ))

R = √(5² + 6² + 2 × 5 × 6 × cos(60°)) = √91 ≈ 9.53 N

2. Resolve a vector of 12 N acting at 45° to the horizontal into its components.

Answer:

Horizontal component: A_x = 12 × cos(45°) ≈ 12 × 0.707 = 8.49 N

Vertical component: A_y = 12 × sin(45°) ≈ 12 × 0.707 = 8.49 N

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