Further Mathematics - Senior Secondary 2 - Vectors in three dimensions

Vectors in three dimensions

Get it on Google Play
Get it on the Apple App Store
  1. Home
  2. Comprehensive Lesson Plan and Notes
  3. Senior Secondary 2
  4. Vectors in three dimensions

TERM: 1ST TERM

WEEK 9
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Vectors in Three Dimensions
Focus: Vector or Cross Product in Three Dimensions, Application of Cross Product

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

  1. Define and understand the concept of the cross product of two vectors in three dimensions.
  2. Solve problems on finding the cross product of two vectors.
  3. Apply the cross product to real-life situations, such as finding the area of a parallelogram and determining the perpendicular vector.

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Practice exercises
  • Real-life application discussions
  • Problem-solving activities

INSTRUCTIONAL MATERIALS:

  • Whiteboard and markers
  • Charts illustrating the cross product formula
  • Graphing paper and rulers for practical exercises
  • Worksheets with practice problems on the cross product and its applications

PERIOD 1 & 2: Introduction to Cross Product in Three Dimensions

PRESENTATION:

               2. Find the cross product of A = (0, -1, 2) and B = (3, 2, 1).

               3. If two vectors are parallel, what is their cross product?

               4. What does the magnitude of the cross product represent?

               5. In what scenarios can we use the cross product in real-life applications?

CLASSWORK (5 questions):

  1. Find the cross product of A = (2, -1, 0) and B = (1, 3, 4).
  2. What is the angle between two perpendicular vectors?
  3. Calculate the cross product of A = (1, 1, 1) and B = (1, 1, -1).
  4. Given two vectors A and B, explain what happens if their cross product is zero.
  5. How would you use the cross product to determine the area of a parallelogram formed by two vectors?

ASSIGNMENT (5 tasks):

  1. Research a real-life application of the cross product and explain it in your own words.
  2. Calculate the cross product of A = (5, 3, 2) and B = (7, 4, 3).
  3. Find the area of a parallelogram formed by two vectors A = (3, 0, 4) and B = (0, 5, 2).
  4. Find the cross product of A = (6, 1, -2) and B = (3, -2, 4).
  5. Discuss how the cross product can be used in physics to find torque.

PERIOD 3 & 4: Application of Cross Product

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Applications

Explains the practical uses of the cross product in real-world scenarios such as calculating the torque and the area of a parallelogram formed by two vectors.

Students listen attentively and note the applications.

Step 2 - Torque Example

Demonstrates how the cross product is used to calculate the torque, which is the rotational effect of a force about a point: τ=r×F Uses an example with given vectors r and F.

Students follow the steps for solving the torque problem and ask questions for clarification.

Step 3 - Guided Practice

Provides problems on applications of the cross product, including torque and area of a parallelogram. Students work on these problems in pairs.

Students work in pairs, discussing their solutions to the given problems.

Step 4 - Practical Applications Discussion

Facilitates a discussion on other uses of the cross product, such as in physics (e.g., in electromagnetism, angular momentum), and in computer graphics for determining normal vectors.

Students engage in the discussion and provide their own insights.

NOTE ON BOARD

    
  • Torque: τ=r×F
  • Area of Parallelogram: Area=∣A×B∣

EVALUATION (5 exercises):

  1. Calculate the torque given r = (1, 2, 3) and F = (4, 5, 6).
  2. Find the area of a parallelogram formed by vectors A = (1, 0, 0) and B = (0, 2, 0).
  3. Calculate the cross product of r = (2, 0, 1) and F = (1, 1, 0) to find the torque.
  4. Solve for the torque when r = (3, 1, 2) and F = (0, -1, 4).
  5. Explain the importance of the cross product in calculating the moment of force.

CLASSWORK (5 questions):

  1. Calculate the area of a parallelogram formed by vectors A = (2, -3, 1) and B = (4, 2, 3).
  2. What is the cross product of vectors A = (0, 1, 2) and B = (1, 2, 3) in terms of magnitude?
  3. Calculate the torque produced by r = (2, 1, 0) and F = (3, 4, 5).
  4. How does the direction of the cross product relate to the right-hand rule?
  5. What happens to the cross product when two vectors are parallel?

ASSIGNMENT (5 tasks):

  1. Research how the cross product is used in computer graphics.
  2. Calculate the torque produced by r = (1, 0, 0) and F = (0, 1, 0).
  3. Find the area of a parallelogram formed by A = (1, 0, 0) and B = (0, 1, 0).
  4. Calculate the cross product of A = (3, 4, 5) and B = (6, 7, 8).
  5. How is the cross product used to determine the normal vector of a surface?