Further Mathematics - Senior Secondary 1 - Vectors in two dimensions II

TERM: 2^ND TERM

WEEK 6

Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Vectors in Two Dimensions II
Focus: Triangle Law of Vectors, Parallelogram Law of Vectors, Resolution of Vectors

SPECIFIC OBJECTIVES:

By the end of the lesson, students should be able to:

Understand and apply the Triangle Law of Vectors.
Understand and apply the Parallelogram Law of Vectors.
Resolve a vector into two perpendicular components (horizontal and vertical directions).
Apply vector resolution in practical situations.

INSTRUCTIONAL TECHNIQUES:

Question and answer
Demonstration
Guided practice
Discussion
Real-life connections and examples
Analogy

INSTRUCTIONAL MATERIALS:

Whiteboard and markers
Charts of resolved vectors
Rulers and protractors for diagram drawing
Worksheets for practice exercises
Vector diagrams for demonstration

PERIOD 1 & 2: Introduction to Triangle Law of Vectors and Parallelogram Law of Vectors

PRESENTATION:

Step	Teacher’s Activity	Student’s Activity
Step 1 - Introduction	Introduces the concept of vectors and their importance in physics and mathematics. Explains how vectors are represented by arrows.	Students listen attentively and ask questions.
Step 2 - Triangle Law of Vectors	Explains the Triangle Law of Vectors: the sum of two vectors can be represented by the third side of a triangle formed by the vectors. Demonstrates with a diagram.	Students observe and take notes.
Step 3 - Parallelogram Law of Vectors	Explains the Parallelogram Law of Vectors: the sum of two vectors can be obtained by completing a parallelogram, where the diagonals represent the sum of the vectors. Demonstrates with a diagram.	Students observe and take notes.
Step 4 - Application	Demonstrates both laws with examples. Shows how to add vectors graphically using both the Triangle and Parallelogram Laws.	Students observe and work through example problems.

NOTE ON BOARD:

Triangle Law of Vectors: The sum of two vectors is represented by the third side of the triangle formed by the vectors.
Parallelogram Law of Vectors: The sum of two vectors is represented by the diagonal of the parallelogram formed by the vectors.

EVALUATION (5 exercises):

State the Triangle Law of Vectors.
State the Parallelogram Law of Vectors.
Draw a triangle showing the application of the Triangle Law of Vectors.
Draw a parallelogram showing the application of the Parallelogram Law of Vectors.
Give an example of where the Parallelogram Law of Vectors is applied in real life.

CLASSWORK (5 questions):

Using the Triangle Law of Vectors, find the resultant of two vectors A\mathbf{A} and B\mathbf{B}.
Using the Parallelogram Law of Vectors, find the resultant of two vectors C\mathbf{C} and D\mathbf{D}.
Describe the geometric construction for adding two vectors using the Triangle Law of Vectors.
Draw a parallelogram to represent two vectors and find their resultant.
If vector {P} is 4 units and vector {Q} is 5 units, use the Parallelogram Law to find their resultant if they are at 90° to each other.

ASSIGNMENT (5 tasks):

Research and describe an application of the Triangle Law of Vectors in engineering.
Use the Parallelogram Law of Vectors to solve for the resultant of two vectors at an acute angle.
Draw a diagram representing the resolution of a vector into its components.
Provide real-life examples where the resolution of vectors is applied.
If two vectors form a right-angled triangle, calculate their resultant using the Triangle Law.

PERIOD 3 & 4: Resolution of Vectors

PRESENTATION:

Step	Teacher’s Activity	Student’s Activity
Step 1 - Introduction to Resolution	Introduces the concept of vector resolution and explains how a vector can be split into two perpendicular components: horizontal and vertical.	Students listen and take notes.
Step 2 - Example of Vector Resolution	Demonstrates resolving a vector (e.g., 5N at 30°) into horizontal and vertical components using trigonometric functions (cosine and sine).	Students observe and follow the example.
Step 3 - Practical Application	Uses real-life examples (e.g., forces acting on an object at an angle) to show the importance of vector resolution.	Students discuss examples and participate in solving problems.
Step 4 - Guided Practice	Provides exercises where students resolve vectors into components using trigonometric methods (e.g., resolving 10N at 60°).	Students practice the resolution of vectors in pairs, seeking assistance when needed.

NOTE ON BOARD:

Resolution of Vectors:

Horizontal Component = Rcos⁡θ
Vertical Component = Rsin⁡θ
Where R is the magnitude of the vector and θ is the angle.

EVALUATION (5 exercises):

Resolve a 10N vector at 30° into horizontal and vertical components.
Resolve a 15N vector at 45° into horizontal and vertical components.
What is the horizontal component of a 20N vector at 60°?
What is the vertical component of a 12N vector at 90°?
Explain the significance of resolving vectors in engineering.

CLASSWORK (5 questions):

Resolve a 25N vector at 40° into horizontal and vertical components.
Resolve a 50N vector at 75° into horizontal and vertical components.
Find the horizontal component of a 30N vector at 20°.
Find the vertical component of a 35N vector at 70°.
If a force of 40N acts at an angle of 30°, resolve it into horizontal and vertical components.

ASSIGNMENT (5 tasks):

Resolve a 60N force acting at 45° into its components.
Given a vector of 10 units at an angle of 90°, resolve it into its components.
Research a practical application of vector resolution in aviation.
Calculate the horizontal and vertical components of a 15N vector at 60°.

Explain the role of vector resolution in the analysis of forces in structures.