Further Mathematics - Senior Secondary 2 - Vectors in three dimensions

Vectors in three dimensions

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TERM: 1ST TERM

WEEK 8
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes (4 periods)
Subject: Further Mathematics
Topic: Vectors in Three Dimensions

  • Scalar Product of Vectors in Three Dimensions
  • Application of Scalar Product

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

  1. Understand the concept of vectors in three dimensions.
  2. Compute the scalar product (dot product) of two vectors in three dimensions.
  3. Apply the scalar product in solving problems related to geometry and physics (e.g., calculating angles between vectors and projection).

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Discussion
  • Practice exercises
  • Analogy and real-life applications

INSTRUCTIONAL MATERIALS:

  • Whiteboard and markers
  • Charts depicting examples of three-dimensional vectors
  • Flashcards with vector problems
  • Graphing tools (optional)

 

PERIOD 1 & 2: Introduction to Vectors in Three Dimensions and Scalar Product

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of vectors in three dimensions, emphasizing that vectors are represented by (x, y, z) coordinates. Gives examples of real-world applications like velocity and force in 3D space.

Students listen attentively and ask questions.

Step 2 - Scalar Product

Introduces the scalar product (dot product) formula for vectors in three dimensions: A·B = A₁B₁ + A₂B₂ + A₃B₃. Provides examples with actual vectors, e.g., A = (1, 2, 3) and B = (4, -5, 6).

Students observe and note down the formula and examples.

Step 3 - Calculation of Scalar Product

Demonstrates how to compute the scalar product of two 3D vectors using the formula. E.g., For A = (1, 2, 3) and B = (4, -5, 6), A·B = 1(4) + 2(-5) + 3(6).

Students follow along and solve examples as the teacher demonstrates.

Step 4 - Real-Life Application

Explains how the scalar product is used to find the angle between two vectors or to compute the projection of one vector onto another. Links to physics concepts such as work done (force × displacement).

Students write down real-life applications and discuss their significance.

NOTE ON BOARD:

  • Vectors in 3D: A = (A₁, A₂, A₃)
  • Scalar product (dot product): A·B = A₁B₁ + A₂B₂ + A₃B₃
  • Example: A = (1, 2, 3), B = (4, -5, 6)
    Scalar Product: A·B = 1×4 + 2×(-5) + 3×6 = 4 - 10 + 18 = 12

EVALUATION (5 exercises):

  1. Find the scalar product of A = (2, 3, 4) and B = (1, 2, 3).
  2. Calculate the scalar product of A = (1, 0, -1) and B = (4, 5, 6).
  3. What is the scalar product of two perpendicular vectors?
  4. How can you use the scalar product to find the angle between two vectors?
  5. Compute the scalar product of A = (0, 0, 1) and B = (1, 1, 1).

CLASSWORK (5 questions):

  1. Find the scalar product of A = (3, 2, 1) and B = (1, 4, 0).
  2. What is the scalar product of A = (1, 1, 1) and B = (-1, -1, -1)?
  3. Calculate the angle between the vectors A = (1, 0, 0) and B = (0, 1, 0) using the scalar product.
  4. What is the scalar product of the vector A = (1, 2, 3) and the zero vector (0, 0, 0)?
  5. Find the projection of A = (2, 3, 4) onto B = (1, 1, 1).

ASSIGNMENT (5 tasks):

  1. Find the scalar product of A = (1, 2, 3) and B = (4, -1, 2).
  2. Calculate the angle between the vectors A = (1, 0, 0) and B = (0, 1, 0).
  3. Compute the projection of A = (3, 4, 0) onto B = (1, 0, 1).
  4. Use the scalar product to determine if the vectors A = (1, 1, 1) and B = (2, 2, 2) are parallel.
  5. Investigate the importance of scalar product in determining the work done by a force.

 

PERIOD 3 & 4: Applications of Scalar Product

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Angle Between Vectors

Explains how the scalar product can be used to find the angle between two vectors using the formula: cos(θ) = (A·B) / (

A

Step 2 - Work Done Calculation

Shows how to use the scalar product to compute work done: Work = Force × Displacement. Gives a real-life example in physics.

Students observe the example and understand how it applies to physics problems.

Step 3 - Guided Practice

Provides problems for students to calculate the angle between vectors, determine projections, and compute work done. Students work in pairs to solve the problems.

Students solve problems under teacher guidance, discussing solutions with peers.

Step 4 - Summary and Conclusion

Summarizes the key concepts of scalar product, its formula, and real-life applications. Reinforces how to use it to calculate angles and projections in physics.

Students ask final questions and clarify doubts.

NOTE ON BOARD:

  • Angle between two vectors: cos(θ) = (A·B) / (|A| |B|)
  • Work done: Work = Force × Displacement (A·B)
  • Example: Given F = (3, 4, 0) and d = (1, 0, 0), Work = 3×1 + 4×0 + 0×0 = 3.

EVALUATION (5 exercises):

  1. Calculate the angle between A = (1, 2, 3) and B = (4, 5, 6).
  2. Determine the work done by a force F = (2, 3, 1) moving an object from position A = (1, 1, 1) to position B = (4, 4, 4).
  3. Find the projection of A = (1, 2, 3) onto B = (4, 5, 6).
  4. Are the vectors A = (1, 2, 3) and B = (-1, -2, -3) parallel?
  5. What is the angle between the vectors A = (1, 0, 0) and B = (0, 1, 0)?

CLASSWORK (5 questions):

  1. Calculate the scalar product of A = (1, 2, 3) and B = (4, 5, 6).
  2. Find the angle between A = (1, 0, 0) and B = (1, 1, 0).
  3. Compute the projection of A = (2, 2, 2) onto B = (1, 1, 1).
  4. Determine the work done by a force F = (2, 1, 0) moving an object from A = (0, 0, 0) to B = (2, 1, 3).
  5. Are the vectors A = (3, 4, 0) and B = (6, 8, 0) parallel?

ASSIGNMENT (5 tasks):

  1. Find the angle between the vectors A = (2, 1, 0) and B = (1, 3, 0).
  2. Calculate the scalar product of A = (2, 3, 4) and B = (-1, 0, 1).
  3. Compute the projection of A = (1, 1, 1) onto B = (2, 2, 2).
  4. Determine if the vectors A = (1, 2, 3) and B = (2, 4, 6) are parallel using the scalar product.
  5. Write a real-life scenario where scalar products are used.