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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:
Vx = V cos(θ)
Vy = 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:
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.
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