Lesson Note on Vectors in Three Dimensions: Cross Product and Applications for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 1st Term
Week: 9
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Algebra and Geometry
Previous Lesson: Vectors in Three Dimensions: Meaning, Scalar Product and Applications.
Topic: VECTORS IN THREE DIMENSIONS
Subject Matter: Vector or cross product in three dimensions, application of cross product

Specific Objectives

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

Cognitive Domain:

• Define the cross product of two vectors in three dimensions.
• State the properties of the cross product.
• Recall the formulas for the area of a parallelogram and for finding a perpendicular vector using the cross product.

Affective Domain:

• Appreciate the importance of the cross product in solving geometrical problems.
• Show interest in solving problems involving vector cross products.

Psychomotor Domain:

• Calculate the cross product of two given vectors using the determinant method.
• Solve problems involving the area of a parallelogram using the cross product.
• Determine a vector perpendicular to two given vectors using the cross product.

Social Domain:

• Collaborate effectively with peers in solving vector cross product problems.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New Further Mathematics for Senior Secondary Schools by P.N. Okeke

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing the determinant method for finding the cross product.
• Whiteboard and markers.

Rationale for the Lesson

This lesson helps pupils understand how to multiply two vectors to get another vector that is perpendicular to both. This knowledge is important for solving real-world problems in physics and engineering, such as calculating torque or magnetic force, and for determining areas of parallelograms in three-dimensional space.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of vectors in two and three dimensions, dot product, and how to evaluate 3×3 determinants.

Lesson Content/Board Summary

VECTORS IN THREE DIMENSIONS: CROSS PRODUCT

1. Definition of Cross Product (Vector Product)

The cross product of two vectors, a and b, denoted by a × b, is a vector that is perpendicular to both a and b. Its magnitude is given by |a||b|sinθ, where θ is the angle between the vectors. The direction is determined by the right-hand rule.

2. Calculating the Cross Product (Determinant Method)

If a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then their cross product is:

a × b =

| i   j   k   |
| a₁  a₂  a₃  |
| b₁  b₂  b₃  |

Expanding the determinant gives:

a × b = i(a₂b₃ – a₃b₂) – j(a₁b₃ – a₃b₁) + k(a₁b₂ – a₂b₁)

Example 1: Find the cross product of a = 2i + 3j + k and b = i – 2j + 4k.

Solution:

Step 1: Set up the determinant.

a × b = | i   j   k |
          | 2   3   1 |
          | 1  -2   4 |

Step 2: Expand the determinant.

= i((3)(4) – (1)(-2)) – j((2)(4) – (1)(1)) + k((2)(-2) – (3)(1))

Step 3: Simplify the terms.

= i(12 – (-2)) – j(8 – 1) + k(-4 – 3)

= i(14) – j(7) + k(-7)

= 14i – 7j – 7k

3. Properties of Cross Product

The following are important properties of the cross product:

• Anticommutative: a × b = – (b × a)
• Cross product of parallel vectors: If a and b are parallel, then a × b = 0. (This implies a × a = 0)
• Distributive property: a × (b + c) = (a × b) + (a × c)
• Scalar multiplication: (ka) × b = k(a × b) = a × (kb)

4. Applications of Cross Product

a) Area of a Parallelogram

If two adjacent sides of a parallelogram are represented by vectors a and b, then the area of the parallelogram is the magnitude of their cross product:

Area = |a × b|

Example 2: Find the area of the parallelogram with adjacent sides given by vectors u = 3i + j – 2k and v = i – 3j + 4k.

Solution:

Step 1: Calculate the cross product u × v.

u × v = | i   j   k |
          | 3   1  -2 |
          | 1  -3   4 |

= i((1)(4) – (-2)(-3)) – j((3)(4) – (-2)(1)) + k((3)(-3) – (1)(1))

= i(4 – 6) – j(12 – (-2)) + k(-9 – 1)

= -2i – 14j – 10k

Step 2: Find the magnitude of the resultant vector.

|u × v| = √((-2)² + (-14)² + (-10)²)

= √(4 + 196 + 100)

= √300

= √(100 × 3)

= 10√3

Step 3: State the area.

Area = 10√3 square units.

b) Finding a Perpendicular Vector

The cross product a × b results in a vector that is perpendicular (orthogonal) to both vector a and vector b.

Example 3: Find a vector perpendicular to both p = i + 2j – k and q = 2i – j + 3k.

Solution:

Step 1: Calculate the cross product p × q.

p × q = | i   j   k |
          | 1   2  -1 |
          | 2  -1   3 |

= i((2)(3) – (-1)(-1)) – j((1)(3) – (-1)(2)) + k((1)(-1) – (2)(2))

= i(6 – 1) – j(3 – (-2)) + k(-1 – 4)

= i(5) – j(5) + k(-5)

= 5i – 5j – 5k

Step 2: State the perpendicular vector.

A vector perpendicular to both p and q is 5i – 5j – 5k.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher reviews the concept of vector dot product and asks pupils to state its result (scalar). The teacher then introduces that today’s lesson will focus on another type of vector multiplication that results in a vector.
Pupils’ Activity: Pupils recall the dot product and its properties, and listen attentively to the introduction of the cross product.
Learning Point: Pupils are prepared for the new topic by linking it to previous knowledge.

Step 2: Definition and Formula of Cross Product

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines the cross product (vector product) of two vectors and explains its geometric meaning (perpendicular vector). The teacher then introduces the determinant method for calculating the cross product of two vectors in three dimensions.
Pupils’ Activity: Pupils write down the definition and the formula for the cross product.
Learning Point: Pupils understand what a cross product is and how it is represented mathematically.

Step 3: Calculating Cross Product (Worked Example)

Time: 10 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher demonstrates how to calculate the cross product of two vectors using a worked example on the board, explaining each step of the determinant expansion.
Pupils’ Activity: Pupils carefully follow the steps demonstrated by the teacher and copy the example into their notes.
Learning Point: Pupils learn the practical method for computing the cross product of vectors.

Step 4: Properties of Cross Product

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains the key properties of the cross product, such as anticommutativity, and the condition for parallel vectors.
Pupils’ Activity: Pupils listen and note down the properties of the cross product.
Learning Point: Pupils understand the fundamental rules governing vector cross products.

Step 5: Application 1: Area of a Parallelogram

Time: 8 minutes
Teaching Skill: Problem Solving/Guidance
Teacher’s Activity: The teacher introduces the application of the cross product in finding the area of a parallelogram. The teacher guides pupils through a worked example on the board, calculating the cross product and then its magnitude to find the area.
Pupils’ Activity: Pupils actively participate in solving the problem and copy the solution into their notes.
Learning Point: Pupils learn to apply the cross product to calculate the area of a parallelogram.

Step 6: Application 2: Finding a Perpendicular Vector

Time: 5 minutes
Teaching Skill: Problem Solving/Guidance
Teacher’s Activity: The teacher explains how the cross product directly yields a vector perpendicular to the two original vectors. The teacher works through an example to illustrate this concept.
Pupils’ Activity: Pupils observe the process and record the example of finding a perpendicular vector.
Learning Point: Pupils understand how to use the cross product to determine a vector orthogonal to two given vectors.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define the cross product of two vectors.
2. State any two properties of the cross product.
3. Given a = i + 2j and b = 3i – k, calculate a × b.
4. How can the cross product be used to find the area of a parallelogram?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teacher’s Activity: The teacher summarizes the key points of the lesson, emphasizing the definition, calculation, and applications of the cross product. The teacher assigns homework for further practice.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: The lesson is reinforced, and pupils are encouraged to practice.

Lesson Keywords

• Cross product – A vector resulting from the multiplication of two vectors, perpendicular to both.
• Vector product – Another term for cross product.
• Determinant – A scalar value calculated from the elements of a square matrix, used in calculating the cross product.
• Parallelogram – A quadrilateral with two pairs of parallel sides, whose area can be found using the cross product of its adjacent sides.
• Perpendicular vector – A vector that forms a 90-degree angle with another vector or plane.
• Right-hand rule – A mnemonic used to determine the direction of the cross product vector.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex problems involving triple scalar or vector products. For struggling pupils, the teacher will offer one-on-one support, provide simpler examples, and use visual aids to reinforce the determinant method for calculating the cross product.

Note for teachers using this lesson plan

Ensure pupils are comfortable with 3×3 determinants before introducing the cross product. Emphasize the geometric interpretation of the cross product as a vector perpendicular to the plane formed by the two original vectors. Encourage pupils to use the right-hand rule to visualize the direction. Provide ample practice problems for both calculation and application, including finding unit vectors perpendicular to two given vectors.