Further Mathematics - Senior Secondary 1 - Vectors in Two Dimensions III - Scalar (Dot) Product and Its Application

Vectors in Two Dimensions III - Scalar (Dot) Product and Its Application

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TERM: 2ND TERM

WEEK 7
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Vectors in Two Dimensions III – Scalar (Dot) Product and Its Application
Focus: Understanding the scalar (dot) product, its formula, and its geometric and trigonometric applications.

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

  1. Define the scalar (dot) product of two vectors.
  2. Apply the scalar product in geometry and trigonometry.
  3. Solve problems involving the scalar product in practical situations.

INSTRUCTIONAL TECHNIQUES:
• Lecture and Explanation
• Guided Demonstration
• Application of formulas through practical examples
• Question and Answer
• Problem-solving exercises

INSTRUCTIONAL MATERIALS:
• Whiteboard and markers
• Charts illustrating geometrical applications of the scalar product
• Vectors’ diagrams showing angles between vectors
• Calculators (optional)

PERIOD 1 & 2: Introduction to Scalar (Dot) Product

PRESENTATION:

EVALUATION (5 Exercises):

  1. Given vectors A=(2,3) and B=(4,−1) calculate A⋅B
  2. What is the scalar product of two perpendicular vectors?
  3. If A=(5,2) and B=(3,6) find the angle between them.
  4. Explain the significance of the scalar product being zero.
  5. Find the scalar product of vectors A=(3,4) and B=(2,1)

CLASSWORK (5 Questions):

  1. Calculate the scalar product of A=(6,7) and B=(3,4)
  2. Given vectors A=(8,−4) and B=(5,3) determine the angle between them.
  3. Find the scalar product of vectors A=(2,5) and B=(6,3)
  4. How does the scalar product change when the two vectors are perpendicular?
  5. Explain why the scalar product can be used to find the angle between vectors.

ASSIGNMENT (5 Tasks):

  1. Derive the formula for the scalar product using the angle between two vectors.
  2. Solve a problem where the scalar product of two vectors is used to find the projection of one vector onto another.
  3. Using vectors A=(1,2) and B=(4,−3) find the angle between the vectors.
  4. Explain the real-life applications of the scalar product in geometry and physics.
  5. Find the scalar product of two vectors in 3D space.

PERIOD 3 & 4: Application of Scalar (Dot) Product in Geometry and Trigonometry

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 – Problem Introduction

Provides problems related to the application of the scalar product in geometry, e.g., finding the projection of one vector onto another or the angle between vectors in a triangle.

Students attempt the problems, asking for clarification if needed.

          5. Solve a problem using the scalar product to calculate the angle between two vectors in 3D space.

CLASSWORK (5 Questions):

  1. Using vectors A=(6,7) and B=(2,5) find the projection of
  2. Solve for the angle between A=(5,6) and B=(7,−3)
  3. Given vectors A=(4,1) and B=(3,−2) calculate the area of the parallelogram they form.
  4. Find the scalar product of vectors A=(2,5) and B=(4,3)
  5. Derive the formula for the projection of one vector onto another.

ASSIGNMENT (5 Tasks):

  1. Solve for the projection of vector A=(3,1) on B=(5,2)).
  2. Find the angle between vectors A=(8,−2) and B=(7,4)
  3. Use the scalar product to calculate the area of a triangle given vectors A=(2,3) and B=(4,5)
  4. Discuss the role of scalar product in finding distances between points in space.
  5. Solve a real-life problem involving the use of the scalar product, such as calculating the angle between two vectors in a physics context.